**Cite This:**

*J. Chem. Theory Comput.*2021, 17, 4, 2166-2185

# Optimizing Molecular Geometries in Strong Magnetic Fields

- Tom J. P. Irons
*****Tom J. P. IronsSchool of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom*****(T.J.P.I.) Email: [email protected]More by Tom J. P. Irons - ,
- Grégoire DavidGrégoire DavidSchool of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD, United KingdomMore by Grégoire David
- , and
- Andrew M. Teale
*****Andrew M. TealeSchool of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD, United KingdomHylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P. O. Box 1033 Blindern, N-0315 Oslo, Norway*****(A.M.T.) Email: [email protected]More by Andrew M. Teale

## Abstract

An efficient implementation of geometrical derivatives at the Hartree–Fock (HF) and current-density functional theory (CDFT) levels is presented for the study of molecular structure in strong magnetic fields. The required integral derivatives are constructed using a hybrid McMurchie–Davidson and Rys quadrature approach, which combines the amenability of the former to the evaluation of derivative integrals with the efficiency of the latter for basis sets with high angular momentum. In addition to its application to evaluating derivatives of four-center integrals, this approach is also applied to gradients using the resolution-of-the-identity approximation, enabling efficient optimization of molecular structure for many-electron systems under a strong magnetic field. The CDFT contributions have been implemented for a wide range of density functionals up to and including the meta-GGA level with current-density dependent contributions and (range-separated) hybrids for the first time. Illustrative applications are presented to the OH and benzene molecules, revealing the rich and complex chemistry induced by the presence of an external magnetic field. Challenges for geometry optimization in strong fields are highlighted, along with the requirement for careful analysis of the resulting electronic structure at each stationary point. The importance of correlation effects is examined by comparison of results at the HF and CDFT levels. The present implementation of molecular gradients at the CDFT level provides a cost-effective approach to the study of molecular structure under strong magnetic fields, opening up many new possibilities for the study of chemistry in this regime.

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### License Summary*

You are free to share (copy and redistribute) this article in any medium or format and to adapt (remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

Creative Commons (CC): This is a Creative Commons license.

Attribution (BY): Credit must be given to the creator.

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

### License Summary*

You are free to share (copy and redistribute) this article in any medium or format and to adapt (remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

Creative Commons (CC): This is a Creative Commons license.

Attribution (BY): Credit must be given to the creator.

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

## 1. Introduction

## 2. Integrals and Derivative Integrals over LAOs

### 2.1. London Atomic Orbitals

**A**= (A

_{x}, A

_{y}, A

_{z}), has angular momentum

**a**= (

*a*

_{x},

*a*

_{y},

*a*

_{z}), and has

*K*

_{a}exponents {α

_{μ}} with respective contraction coefficients {

*d*

_{μ}}. LAOs are a generalized form of GTOs, (29) comprising the form in eq 1 multiplied by a complex phase factor,

**k**

_{a}is the wave vector of the London plane wave , depending on the external magnetic field and the position of the LAO relative to the gauge-origin

**O**. In the limit of , the LAO in eq 2 will reduce to the corresponding GTO in eq 1. As described in ref (15), the basis may be transformed from the Cartesian representation to the solid harmonic representation with LAOs in the same way as for GTOs, with coefficients constructed explicitly with the method of Schlegel and Frisch (31) or alternatively by recurrence relation as described in ref (32).

### 2.2. Generalized Shell Pairs

_{a}(

**r**) and ω

_{b}(

**r**) represent charge distributions,

**a**

_{μ}

**b**

_{ν}| represents the product of the μth and νth individual contractions of ω

_{a}and ω

_{b}, respectively, while (

**ab**| is the overall inner product of the two LAOs; if both are primitive, the two definitions are equivalent.

*s*-type LAOs, centered on

**A**and

**B**, with exponents α and β, contraction coefficients

*d*

_{a}and

*d*

_{b}, and phase factors

**k**

_{a}and

**k**

_{b}, respectively, the product may be written as

*s*-type LAOs in eq 4 can be written more concisely as

*Ũ*

_{P}| ≤ 10

^{–12}, the pair of primitive functions may be considered negligible and discarded from the shell-pair; this allows an increasingly large proportion of the Gaussian product space to be discarded as the system becomes larger. Within this framework of (reduced) shell-pairs, the contraction of eq 3 may be applied as early as possible in each integral algorithm to yield contracted integrals.

### 2.3. London Hermite Gaussian Functions

**p**|. The use of Hermite Gaussian functions as intermediates in integral evaluation can provide a computational advantage since, according to the Leibniz theorem, the differential operators over the nuclear coordinates/Gaussian product center can be moved outside of the integral over electronic coordinates. Additionally, higher-order Hermite Gaussian functions can be obtained from lower-order Hermite Gaussian functions by a simple recurrence relation derived as

*x*– P

_{x}) may be moved in front of the differential by using the relation

**ab**|

**p**] = 0 if

**p**< 0 or

**p**>

**a**+

**b**. The transformation between Hermite and Cartesian Gaussian is thus seen to be (i) not explicitly dependent on

**P**and (ii) independent of the London phase factor. For convenience in later discussions and consistent with eq 5,

*U*

_{P}will be included in the Hermite Gaussian prefactor, yielding the modified definitions

## 3. Molecular Integrals

### 3.1. Overlap and Kinetic Energy Integrals

**P**̃ from eq 5, the Obara–Saika recurrence relation for the overlap of two primitive LAOs can be summarized as

*n*th-order multipole and differential operators respectively as

*x*-component of the kinetic energy integral may be written as the sum of mixed multipole-moment integrals obtained straightforwardly from overlap integrals described in ref (15),

### 3.2. Nuclear Attraction Integrals

**C**is the position of an atomic nucleus with unit charge. In the present work, the Hermite Gaussian expansion described in Section 2.3 will be substituted directly into eq 22, assuming primitive LAOs, as the first step to deriving a method for its evaluation,

*s*-type Gaussian function, centered in the complex plane, to be integrated over the Coulomb operator. The Coulomb operator may then be eliminated by substituting it with its Laplace transform,

*m*th-order molecular incomplete γ function, or the Boys Function. (46) These functions cannot be evaluated analytically so must be approximated numerically; methods by which this may be done are described elsewhere. (1,27,43,47−49) For convenience, the following intermediate function is defined, combining the Boys Function with a prefactor by which it is always multiplied

### 3.3. Electron Repulsion Integrals

*Ũ*

_{Q}, and

**Q**̃ are the second shell-pair equivalents of ζ,

*Ũ*

_{P}, and

**P̃**, respectively. To derive the integral expression, it is first necessary to substitute the Hermite Gaussian expansion of a primitive shell-pair into eq 37,

*n*th derivative of a product to yield

## 4. Derivative Integrals

*x*-,

*y*-, and

*z*-components of each Gaussian center. The complexity increases further when considering the derivatives of LAOs, since differentiation of the phase-factor results in additional terms not otherwise present.

### 4.1. Overlap and Kinetic Energy Integral Derivatives

*s*-type LAOs and here generalized to higher angular momentum as

*x*-coordinate of each Gaussian function in turn can be written as

_{y}and π̂

_{z}components, to yield the full derivative integral which may be contracted and transformed to spherical harmonics as appropriate.

### 4.2. Nuclear Attraction Integral Derivatives

### 4.3. Electron Repulsion Integral Derivatives

### 4.4. Modified Approach to Constructing LAO Derivative ERIs

*w*

_{λ}as (15,30,53,54)

*t*

_{λ}

^{2}are the roots of the Rys polynomial. The resolution of the integrand into Cartesian components allows angular momentum to be incremented separately in each direction, resulting in a vertical recurrence relation (VRR) that scales more favorably with angular momentum than that in eq 42 (or indeed the Head-Gordon–Pople recurrence relation, (55) discussed in ref (15)). Hermite integrals are obtained by multiplying the relevant

*x*-,

*y*-, and

*z*-components of the 2D integrals and summing over the Rys polynomial nodes,

*N*is the number of Rys quadrature points; for the case of the integral derivative,

*N*= (

*L*

_{p}+

*L*

_{q}+ 2)/2. This summation step is generally the computational bottleneck of the Rys quadrature approach, scaling less favorably with angular momentum than the comparatively inexpensive VRR.

**p**|

**q**] integrals, each combination of

*x*- and

*y*-components is frequently combined with multiple

*z*-components; thus, creation of an

*xy*-intermediate to be combined with many

*z*-components in summation over Rys quadrature nodes can reduce the number of individual multiplications required by the number of quadrature points for each reuse of the intermediate. Additionally, unnecessary multiplication by unity is avoided by discarding and from summations where these occur; cannot be discarded as it carries the Rys weights and other prefactors.

## 5. Hartree–Fock Gradients

*x*-derivative of the one- and two-electron integrals with respect to a given nucleus N can be written as (58)

*D*

_{ab}is the density matrix, given by the sum of the spin-density matrices,

*D*

_{ab}=

*D*

_{ab}

^{α}+

*D*

_{ab}

^{β}, and

*W*

_{ab}is the energy-weighted density matrix, constructed from the spin-density and Fock matrices

*F*

_{ab}

^{σ}as

### 5.1. Analytical Gradients with the Resolution-of-the-Identity Approximation

**P**,

**Q**); four-center integrals are approximated by a contraction of three-center integrals with the inverse of the Coulomb metric of the auxiliary functions

## 6. Analytical Gradients with Current Density Functional Theory

*E*

_{xc}is typically approximated at each point in space by some function

*f*of local or semi-local quantities

_{σ}as is the case for most meta-GGAs, dependence on the paramagnetic current density

**j**

_{pσ}is required to ensure the xc energy is invariant with respect to gauge transformation. (63−66) These quantities, in addition to the electron density ρ

_{σ}and its derivatives can be evaluated from the basis of LAOs in which the Kohn–Sham one-electron orbitals are expanded and the spin density matrix as

### 6.1. Matrix Elements of the XC Potential

**ξ**as in eq 74, can be written as (67)

*f*cannot be written in the same way since these are implicit and not explicit functionals of the density.

### 6.2. XC Contribution to Nuclear Gradient

**ξ**, the gradient of the xc energy with respect to nuclear position is defined as (67)

## 7. Results and Discussion

_{0}are known to exist on the surfaces of magnetic white dwarf stars, (18−20) for which the atmospheres are often dominated by hydrogen and helium but are thought to be abundant in many other elements. (69−71) Many studies have examined the effects of strong magnetic fields on atomic energy levels, (72−75) essential for interpreting the spectra observed from these stellar objects. Astrochemical observations also suggest that simple diatomic molecules exist in the atmospheres of magnetic white dwarf stars. (21,76,77) Modeling the effects of strong magnetic fields on the spectra of molecules has been much more limited than the studies on atoms, (78,79) although they have become the subject of increasing interest more recently. (5,14) While molecular studies have become more common, the effects of magnetic field strength on molecular geometry have had much less consideration. (80) As such, the first system we consider will be the OH diatomic molecule, the properties of which in strong magnetic fields have been the subject of recent astrochemical interest. (81) In particular, we will see that the behavior of this small system under the strong magnetic fields considered here is well-explained by consideration of orbital paramagnetic interactions with the field.

### 7.1. Computational Details

^{–4}au, the root-mean-square of the gradient and of the ensuing step <2 × 10

^{–4}au, and the change in energy between steps <5 × 10

^{–6}au.

*u*-aug-cc-pCVTZ while for benzene it was

*u*-aug-cc-pVDZ; these two molecules were considered in magnetic fields up to 0.20B

_{0}and 0.15B

_{0}, respectively (B

_{0}= ℏ

*e*

^{–1}

*a*

_{0}

^{–2}= 2.3505 × 10

^{5}T), in which ranges the basis sets selected should provide an adequate description of field-induced density changes. (14,98)

*u*-aug-cc-pVDZ-RI basis used as the auxiliary basis; (99) recent work has shown that RI may be used with LAOs in a similar way to its use with standard GAOs. (16,27,30)

### 7.2. Equilibrium Geometry of OH

*M*

_{s}= −

^{1}/

_{2}in strong magnetic fields is investigated. For this system, the potential energy curve can be correctly represented from equilibrium to dissociation using a single determinant. However, even for this simple molecule, the presence of a magnetic field significantly complicates the potential energy surface, with consideration of the underlying physics required to interpret the optimized geometries obtained.

_{∞h}. In general, only rotation axes parallel to the field, mirror planes perpendicular to the field, and the center of inversion, if present, will remain. (102)

_{2}induced by a strong magnetic field applied perpendicular to the internuclear axis. (103) A more general analysis of these phenomena has recently been presented by Austad et al. in ref (104).

*z*-axis, for which the Hamiltonian can be written as

*ŝ*

_{z}the spin angular momentum operator, and

*l̂*

_{z}the orbital angular momentum operator. These terms result in the spin–Zeeman and orbital paramagnetic contributions to the energy in a field, respectively, while the final term yields the diamagnetic contribution to the energy. The spin- and angular momentum-dependent terms can cause an increase or decrease in the energy with respect to field strength, whereas the diamagnetic term will always result in an increase in the energy with field strength; due to its quadratic dependence on , it will always become the dominant term at sufficiently high field strengths.

_{∞v}and its electronic ground state and first excited state have the electronic configurations

*s*and 2

*s*orbitals are omitted from the electronic configuration of the oxygen atoms; however, they are occupied as 1

*s*

^{αβ}2

*s*

^{αβ}in all cases, and this is assumed throughout.

^{2}Π state of OH is the lowest in energy for all bond lengths computed, from equilibrium to near-dissociation. Furthermore, the only stationary points along the potential energy curve are at equilibrium and in the dissociation limit; hence, the equilibrium geometry is located easily from different starting geometries. We will consider optimizations starting from 1.6 and 3.2 au here for different field strengths and orientations.

#### 7.2.1. OH in a Magnetic Field Parallel to the Bond

_{v}mirror planes have normals perpendicular to the field and, hence, no longer describe the symmetry of the system in the field. The infinite-order axis of proper rotation along the internuclear axis,

*C*

_{∞}

^{ϕ}, however remains; thus, the point group of the OH molecule in a field parallel to the internuclear axis becomes C

_{∞}. The symmetries by which the molecular orbitals in eqs 91 and 92 are characterized remain the same in this case, with the only significant difference being the reduction in symmetry of the excited state from |

^{2}Σ

^{+}⟩ → |

^{2}Σ⟩.

*M*

_{s}= −

^{1}/

_{2}, the difference between eq 96 and eq 97 arises due to the difference in the orbital paramagnetic contribution to the energy. It would be therefore expected that the |

^{2}Π⟩ state would remain the ground state with increasing field strength parallel to the internuclear axis. Geometry optimizations should then be expected to track the change in equilibrium structure of this state as a function of the magnetic field strength.

_{0}and 0.2B

_{0}applied parallel to the internuclear axis. In this orientation, the energy is stationary with respect to the angle between the internuclear axis and the field; thus, without perturbation, the molecule remains in this alignment throughout the optimization. The equilibrium bond length and the respective binding energy (

*E*

_{bind}=

*E*

_{OH}–

*E*

_{O}–

*E*

_{H}) for OH in this series of magnetic fields are summarized in Table 2.

B/B_{0} | R_{eq}^{HF} | R_{eq}^{TPSS} | E_{bind}^{HF} | E_{bind}^{TPSS} |
---|---|---|---|---|

0.00 | 1.7974 | 1.8564 | –0.10562 | –0.17007 |

0.05 | 1.7967 | 1.8546 | –0.10982 | –0.17288 |

0.10 | 1.7954 | 1.8530 | –0.11058 | –0.17362 |

0.15 | 1.7932 | 1.8504 | –0.11182 | –0.17484 |

0.20 | 1.7902 | 1.8469 | –0.11348 | –0.17645 |

^{a}

Bond lengths are in bohr and binding energies in hartree.

^{2}Π⟩ state of OH in field strengths over the range 0.0–0.2B

_{0}with the internuclear axis aligned parallel to the field. Individual states were tracked along the potential energy curve using the maximum overlap method, generalized for use with complex orbitals. (16,109−112) These are shown at field strengths of 0.1B

_{0}and 0.2B

_{0}in Figures 2 and 3, respectively.

#### 7.2.2. Dissociation Limit

^{2}Π⟩ state at zero field was identified as an O atom with

*M*

_{s}= −1 and the specific configuration O(2

*p*

_{–1}

^{β}2

*p*

_{0}

^{αβ}2

*p*

_{+1}

^{β}). For single determinant models such as HF and CDFT this configuration has lower energy than, for example, the O(2

*p*

_{–1}

^{αβ}2

*p*

_{0}

^{β}2

*p*

_{+1}

^{β}) and O(2

*p*

_{–1}

^{β}2

*p*

_{0}

^{β}2

*p*

_{+1}

^{αβ}) configurations. This is a manifestation of the well-known multiplet problem for these methods—where the configurations contributing to the

^{3}P state of the oxygen atom are not degenerate at zero field. (66,113) By convention, quantities such as binding energies and atomization energies are calculated using the lowest energy configuration predicted by a given theory and this practice has been adopted in calculating the values of binding energy in Table 2.

^{2}Π⟩ state, the dissociation products given in eq 93 will have an initial variation in energy with field strength as

*p*

_{–1}

^{αβ}2

*p*

_{0}

^{β}2

*p*

_{+1}

^{β}) + H(1

*s*

^{α}) vary as

*M*

_{s}= −1 are shown as a function of field strength in Figure 4. It can be seen in Figure 4 that the change in the lowest-energy configuration of oxygen with

*M*

_{s}= −1 occurs at a field strength of 0.027B

_{0}(compared to 0.008B

_{0}with HF, shown in Figure S3). In the binding energies presented in Table 2, it is assumed that the dissociation products at each field strength contain the lowest-energy component of the multiplet; this is confirmed in Figures 2 and 3, which show that the |

^{2}Π⟩ state dissociates into the lower-energy

*M*

_{s}= −1 configuration of oxygen at those field strengths. We note that the multiplet problem for atomic species in magnetic fields has been previously observed by Ivanov and Schmelcher in, for example, refs (73) and (74).

^{2}Σ⟩ state, which are much higher in energy than those of the |

^{2}Π⟩ state at zero field. However, they exhibit a more strongly paramagnetic behavior and the most rapid initial decrease in energy with magnetic field strength,

*M*

_{s}= −

^{1}/

_{2}, the differences in eqs 98–100 are due to the orbital paramagnetic interaction with the field. Accordingly, these dissociation products become the lowest in energy at a field strength of 0.146B

_{0}(compared to 0.165B

_{0}with HF, shown in Figure S3). This is reflected in Figure 3, which shows the potential energy curve of the |

^{2}Π⟩ state tending towards the higher of the two dissociation products. However, the equilibrium geometry of the ground state is correctly located in geometry optimization initialized from bond lengths of both 1.6 and 3.2 au, parallel to the field. This would suggest that the |

^{2}Π⟩ state is not crossed by another, tending to the lower-energy asymptote, at an internuclear distance of 3.2 au or less.

#### 7.2.3. OH in a Magnetic Field Perpendicular to the Bond

*C*

_{∞}

^{ϕ}axis is no longer a symmetry element. However, a mirror plane perpendicular to the magnetic field, in the plane of the internuclear axis does remain; the point group of the OH molecule with a magnetic field perpendicular to the internuclear axis thus becomes C

_{s}. This point group has only two irreducible representations,

*A*′ and

*A*″; the doubly degenerate Π irreducible representation of the zero-field C

_{∞v}point group corresponds to a linear combination of the

*A*′ and

*A*″ irreducible representations in the C

_{s}point group: Π →

*A*′ +

*A*″.

^{2}Π⟩ state upon application of a perpendicular magnetic field reveals their electronic configurations to be

^{2}

*A*′⟩ state dissociates into O(2

*p*

_{–1}

^{αβ}2

*p*

_{0}

^{αβ}) + H(1

*s*

^{β}), whereas the |

^{2}

*A*″⟩ state dissociates into O(2

*p*

_{–1}

^{αβ}2

*p*

_{0}

^{β}2

*p*

_{+1}

^{β}) + H(1

*s*

^{α}). As shown in Figure 4 and discussed in Section 7.2.2, the first of these dissociation products drops in energy below the latter at field strengths of around 0.15–0.16B

_{0}.

^{2}

*A*′⟩ and |

^{2}

*A*″⟩ states of OH in magnetic fields of strengths 0.1B

_{0}and 0.2B

_{0}, respectively, oriented perpendicular to the internuclear axis. In addition, the equilibrium geometry obtained by geometry optimization from initial bond lengths of 1.6 and 3.2 au perpendicular to the field are plotted. It can be seen in both Figures 2 and 3 that the energy of the molecule at equilibrium is lower when aligned parallel to the field than perpendicular; however, the energy of the perpendicular orientation is stationary with respect to rotation relative to the field. The symmetry in this orientation is comparatively high since, upon rotation relative to the field, the system would lose its symmetry with respect to the plane perpendicular to the field and would be reduced to the C

_{1}point group. Therefore, the geometry may be optimized perpendicular to the field if the initial geometry has this orientation.

_{0}, shown in Figure 2, the energy of the |

^{2}

*A*′⟩ state in the perpendicular orientation is lowest at equilibrium but crosses the |

^{2}

*A*″⟩ state at a bond length of around 3.31 au. Therefore, geometry optimization with an initial bond length of 3.2 au tracks the |

^{2}

*A*′⟩ state and correctly identifies its equilibrium geometry, as is the case with an initial bond length of 1.6 au.

_{0}perpendicular to the internuclear axis, the dissociation products of |

^{2}

*A*′⟩ are lower in energy than those of |

^{2}

*A*″⟩; however, the ordering of energies of the states at equilibrium is the same as that at 0.1B

_{0}. Therefore, there is no crossing of these two states along the potential energy curve as there is at 0.1B

_{0}. The potential energy curves for OH in a magnetic field of 0.2B

_{0}perpendicular to the internuclear axis are shown in Figure 3; the equilibrium geometry of the |

^{2}

*A*′⟩ state is correctly located from initial bond lengths of both 1.6 and 3.2 au in these conditions.

_{0}the |

^{2}

*A*′⟩ and |

^{2}

*A*″⟩ states have similar energies at equilibrium and cross as the bond is stretched, so analysis is essential to ascertain which state is obtained in the geometry optimization. To facilitate this assignment, the consideration of the symmetry of the system in the presence of a magnetic field is invaluable. Despite this complexity the geometry optimization using analytic derivatives is able to efficiently locate all of the expected minima, confirming its utility for studying molecular structure and bonding in strong magnetic fields.

### 7.3. Ground-State Structure of Benzene

*M*

_{s}= 0 state (the zero-field ground state) exhibits a shortening of the C–C bonds and extension of the C–H bonds in the presence of a magnetic field of 0.1B

_{0}perpendicular to the molecular plane. Using the present implementation, which allows for unrestricted HF and CDFT optimizations, we investigate the behavior of not only the

*M*

_{s}= 0 state but also the

*M*

_{s}= −1, −2, and −3 states, in which two, four, and six of the π electrons are unpaired, respectively, as a function of magnetic field strength. In each case, the energy as a function of field strength is plotted for the optimized geometries in Figure 5.

*M*

_{s}= 0 state has an energy that rises diamagnetically. Consideration of the

*M*

_{s}= −1, −2, and −3 states highlights the importance of the spin–Zeeman effect in driving progressive unpairing of the π-electrons with increasing field strengths. For HF, the

*M*

_{s}= −1 state decreases in energy with field strength and becomes the ground state at 0.067B

_{0}. For states of higher spin projection, the decrease in energy with field strength is greater due to a larger spin–Zeeman effect; the

*M*

_{s}= −2 state becomes the ground state at 0.089B

_{0}, and the

*M*

_{s}= −3 state becomes the ground state at 0.099B

_{0}. All of the states considered become the ground state at |

**B**| < 0.1B

_{0}.

*M*

_{s}state becomes the ground state. In particular, the

*M*

_{s}= −1 state is the ground state over a much wider range of field strengths compared with that predicted by HF; this is principally due to the greater stabilization of the

*M*

_{s}= −1 state for cTPSS relative to HF. The

*M*

_{s}= −1 state becomes the ground state at 0.075B

_{0}, while the

*M*

_{s}= −2 state becomes the ground state at 0.129B

_{0}and the

*M*

_{s}= −3 state becomes the ground state at 0.137B

_{0}.

**B**| = 0.1B

_{0}for each

*M*

_{s}state. The structures obtained at the cTPSS level are qualitatively similar to those obtained at the HF level (see Figure S4 in the Supporting Information). Geometry optimization in a field determines not only the structural parameters of the molecule, such as bond lengths and angles, but also the preferred orientation of the molecule relative to the external magnetic field.

*M*

_{s}= 0 state has the familiar regular hexagonal arrangement of carbon atoms, with the plane of the molecule oriented perpendicular to the field. The point group of the nuclear framework is D

_{6h}, while the point group of the electronic structure in a magnetic field perpendicular to the plane of the molecule is C

_{6h}. For the

*M*

_{s}= −1 state, the π-system is disrupted by uncoupling two electrons, and as a result there are two unique C–C bond lengths: two of the C–C bonds (in the 1,4 configuration) are longer than the other four, forming an irregular hexagon. The zero-field molecular point group of this structure is D

_{2h}, which is reduced to C

_{2h}in the magnetic field, to which it remains energetically favorable for the molecule to be oriented perpendicular.

*M*

_{s}= −2 state exhibits further disruption of the π-system since four electrons have been unpaired. The zero-field point group is C

_{s}, and the structure may be characterized as a half-chair structure. In contrast to the other spin projections, the molecule in the

*M*

_{s}= −2 state is oriented with the surface area perpendicular to the magnetic field minimized. As the field strength is increased, the orientation evolves such that the mirror plane in the molecular structure is increasingly parallel to the magnetic field. However, since this alignment does not become exact in the range of fields considered here, the overall point group of the structure is reduced to C

_{1}in a magnetic field.

*M*

_{s}= −3 state, all six of the π-electrons have been unpaired and there is no longer a π-system present; the optimized structure adopts a chairlike conformation. At zero field the point group of this structure would be D

_{3d}. In the presence of a magnetic field, the molecule is oriented such that its surface area perpendicular to the field is maximized; the principal axis returns to alignment with the field, with the result that the overall point group of the molecule in the field is S

_{6}.

*M*

_{s}, there are significant differences in quantitative values of the structural parameters such as the C–C and C–H bond lengths. Of particular interest is the variation of the optimized C–C and C–H bond lengths in the

*M*

_{s}= −1 state of benzene with magnetic field strength, shown in Figure 7. There are two sets of lines on each graph since there are two unique C–C and C–H bond lengths in the irregular hexagonal geometry of benzene; the dashed lines represent those for which there are two bonds of that length, whereas the solid lines represent those for which there are four bonds of that length. For this

*M*

_{s}state, the variation of the C–C and C–H bond lengths is significantly different for HF and cTPSS over the range of fields considered. In particular, in Figure 7 it can be seen that the cTPSS structure transitions from an irregular hexagon below ∼0.05

*B*

_{0}to a regular hexagon at higher field strengths.

*M*

_{s}= 0, −2, and −3 the variations of the C–C and C–H bond lengths are qualitatively similar at the HF and cTPSS levels. For the

*M*

_{s}= 0 state of benzene, the HF bond lengths are predicted to be consistently shorter than those from cTPSS. However, the variations of the C–C and C–H bond lengths with increasing field strength follow a similar trend in both cases, with C–C bond lengths decreasing and C–H bond lengths increasing; this is consistent with the behavior observed by Caputo and Lazzeretti (114) and Tellgren et al. (3) These are shown in Figure S5 of the Supporting Information.

*M*

_{s}= −2 state, the low-symmetry C

_{1}structure and rotation of the structure with changing field strength means that there is little further information that can be obtained from this analysis. For the

*M*

_{s}= −3 state, however, the higher-symmetry S

_{6}structure has all equivalent C–C bonds and all equivalent C–H bonds. The C–H bond lengths decrease monotonically with increasing magnetic field strength; the HF bond lengths are consistently shorter than those from cTPSS. The C–C bond lengths show a more complex behavior, first decreasing with field strength before beginning to increase at higher fields, with the HF bond lengths consistently longer than those from cTPSS; this can be seen in Figure S6 of the Supporting Information. The complex behavior of the bonding in this state is interesting and the development of tools for analysis of chemical bonding in strong magnetic fields a focus for future work.

## 8. Conclusions

*M*

_{s}= 0, through a distorted hexagon with

*M*

_{s}= −1, to a half-chair conformation with

*M*

_{s}= −2 at intermediate fields, before adopting a chairlike structure with

*M*

_{s}= −3 at higher fields. These structures reflect the disruption of the π-system as it becomes more and more favorable to unpair electrons in stronger fields. While HF and cTPSS calculations revealed a similar qualitative picture, their comparison showed that the inclusion of correlation can have a significant effect on the predictions of the field strengths at which each state becomes the ground state. A detailed analysis of the bonding in each structure was presented, extending over previous analysis in the literature for the

*M*

_{s}= 0 state. (3,114)

*ab initio*molecular dynamics in strong fields, and coupling of these approaches to real-time electronic structure methods under these conditions. Enabled by the developments presented here, these topics are the focus of ongoing investigation, the results of which will be presented in future work.

## Supporting Information

SI The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.0c01297.

Potential energy curves and optimized geometries of OH in 0.1B

_{0}and 0.2B_{0}magnetic fields parallel and perpendicular to the bond, computed with Hartree−Fock; plot of the energies of OH dissociation products with varying field strength computed with Hartree−Fock; optimized structures of benzene with*M*_{s}= 0, −1, −2, and −3 in a 0.1B_{0}magnetic field computed with Hartree−Fock; plots of optimized C−C and C−H bond lengths in benzene with*M*_{s}= 0 and −3 with varying field strength (PDF)

## Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

## Acknowledgments

We acknowledge financial support from the European Research Council under H2020/ERC Consolidator Grant top DFT (Grant No. 772259). We also acknowledge support from the Engineering and Physical Sciences Research Council (EPSRC), Grant No. EP/M029131/1. We are grateful for access to the University of Nottingham’s Augusta HPC service. A.M.T. has also been supported by the Royal Society University Research Fellowship scheme.

## References

This article references 117 other publications.

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For the largest mols. considered here, the acepleiadylene dianion and the corannulene dianion, the transition field is of the order of 103 T, about one order of magnitude larger than the magnetic field strength currently achievable in exptl. settings. However, our simple model suggests that the paramagnetic-to-diamagnetic transition is a universal property of paramagnetic closed-shell systems in strong magnetic fields, provided no singlet-triplet level crossing occurs for fields smaller than the crit. transition field. Accordingly, fields weaker than 100 T should suffice to trigger the predicted transition for systems whose size is still well within the (medium-large) mol. domain, such as hypothetical antiarom. rings with less than one hundred carbon atoms.**3**Tellgren, E. I.; Reine, S. S.; Helgaker, T. Analytical GIAO and hybrid-basis integral derivatives: application to geometry optimization of molecules in strong magnetic fields.*Phys. Chem. Chem. 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As an illustration of the method, we report a very simple implementation of Hartree-Fock level geometrical derivs. in finite magnetic fields for gauge-origin independent AOs, within the London program. As a quantum-chem. application, we optimize the structure of helium clusters and some well-known covalently bound mols. (water, ammonia and benzene) subject to strong magnetic fields.**4**Tellgren, E. I.; Teale, A. M.; Furness, J. W.; Lange, K. K.; Ekström, U.; Helgaker, T. Non-perturbative calculation of molecular magnetic properties within current-density functional theory.*J. Chem. Phys.*2014,*140*, 034101, DOI: 10.1063/1.4861427Google Scholar4https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXps1emtw%253D%253D&md5=76d696a52c036086a2ad413ec0ee722bNon-perturbative calculation of molecular magnetic properties within current-density functional theoryTellgren, E. I.; Teale, A. M.; Furness, J. W.; Lange, K. K.; Ekstroem, U.; Helgaker, T.Journal of Chemical Physics (2014), 140 (3), 034101/1-034101/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a novel implementation of Kohn-Sham d.-functional theory using London AOs as basis functions. External magnetic fields are treated nonperturbatively, which enable the study of both magnetic response properties and the effects of strong fields, using either std. d. functionals or current-d. functionals-the implementation is the 1st fully self-consistent implementation of the latter for mols. Pilot applications are presented for the finite-field calcn. of mol. magnetizabilities, hypermagnetizabilities, and NMR shielding consts., focusing on the impact of current-d. functionals on the accuracy of the results. Existing current-d. functionals based on the gauge-invariant vorticity are tested and are sensitive to numerical details of their implementation. Also, when appropriately regularized, the resulting magnetic properties show no improvement over std. d.-functional results. An advantage of the present implementation is the ability to apply d.-functional theory to mols. in very strong magnetic fields, where the perturbative approach breaks down. Comparison with high accuracy full-configuration-interaction results show that the inadequacies of current-d. approxns. are exacerbated with increasing magnetic field strength. Std. d.-functionals remain well behaved but fail to deliver high accuracy. The need for improved current-dependent d.-functionals, and how they may be tested using the presented implementation, is discussed in light of the findings. (c) 2014 American Institute of Physics.**5**Sen, S.; Lange, K. K.; Tellgren, E. I. Excited States of Molecules in Strong Uniform and Nonuniform Magnetic Fields.*J. Chem. 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For uniform fields, where comparison is possible, the spectra are found to be qual. similar to those recently obtained with equation of motion coupled cluster theory. We also study the behavior of spin-forbidden excitations with progressive loss of spin symmetry induced by nonuniform magnetic fields. Finally, the equivalence of length and velocity gauges for oscillator strengths when using complex orbitals is investigated and found to hold numerically.**6**Sun, S.; Williams-Young, D. B.; Stetina, T. F.; Li, X. Generalized Hartree–Fock with Nonperturbative Treatment of Strong Magnetic Fields: Application to Molecular Spin Phase Transitions.*J. Chem. 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A Pauli matrix representation of the C-GHF method is introduced to sep. spin contributions from the scalar part of the Fock matrix. Next, spin phase transitions in two different mol. systems are investigated in the presence of a strong magnetic field. Noncollinear spin configurations are obsd. during the spin phase transitions in H2 and a dichromium complex, [(H3N)4Cr(OH)2Cr(NH3)4]4+, with an increase in magnetic field strength. The competing driving forces of exchange coupling and the spin Zeeman effect have been shown to govern the spin phase transition and its transition rate. Addnl., the energetic contributions of the spin Zeeman, orbital Zeeman, and diamagnetic terms to the potential energy surface are also analyzed.**7**Sun, S.; Williams-Young, D.; Li, X. An ab Initio Linear Response Method for Computing Magnetic Circular Dichroism Spectra with Nonperturbative Treatment of Magnetic Field.*J. Chem. Theory Comput.*2019,*15*, 3162– 3169, DOI: 10.1021/acs.jctc.9b00095Google Scholar7https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXms1Wrsbw%253D&md5=ad8cc260678c2169d056bbb8d164a7beAn ab Initio Linear Response Method for Computing Magnetic Circular Dichroism Spectra with Nonperturbative Treatment of Magnetic FieldSun, Shichao; Williams-Young, David; Li, XiaosongJournal of Chemical Theory and Computation (2019), 15 (5), 3162-3169CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Magnetic CD (MCD) expts. provide a sensitive tool for exploring geometric, magnetic, and electronic properties of chem. complexes and condensed matter systems. They are also challenging to simulate because of the need to simultaneously treat the perturbations of a finite magnetic field as well as an optical field. In this work, we introduce an ab initio approach that treats the external magnetic field nonperturbatively with London orbitals for simulating the MCD spectra of closed-shell systems. Effects of a magnetic field are included variationally in the spin-free nonrelativistic Hamiltonian, followed by a linear response formalism to directly calc. the difference in absorption between the left and right circularly polarized light. In addn. to the presentation of underlying math. formalism and implementation, the method developed in this paper has been applied to simulations of MCD spectra of the sodium anion, 2,2,6,6-tetramethylcyclohexanone, and 3-methyl-2-hexanone. Results are discussed and compared to expts.**8**Sun, S.; Beck, R. A.; Williams-Young, D.; Li, X. Simulating Magnetic Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory in Gauge Including Atomic Orbitals.*J. Chem. 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Real-time dynamics are used widely in the simulation of electronic spectroscopies such as absorption as well as electronic CD, but simulating MCD with real-time dynamics is tech. and theor. challenging. In this work, we introduce a real-time dynamics based ab initio method with a non-perturbative treatment of a static magnetic field with London orbitals for simulating the MCD spectra of closed-shell systems. Effects of a magnetic field are included variationally in the spin-free non-relativistic Hamiltonian. Real-time time dependent d. functional theory dynamics are then performed, from which we compute the response function in the presence of the external magnetic field, giving the MCD spectrum. The method developed in this paper is applied to simulate the MCD spectra for pyrimidine, pyrazine, and 1,4-naphthoquinone. Results are discussed and compared to expt.**9**Stopkowicz, S.; Gauss, J.; Lange, K. K.; Tellgren, E. I.; Helgaker, T. 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For this reason, an implementation of a complex CC code is required together with the use of gauge-including AOs to ensure gauge-origin independence. Results of coupled-cluster singles-doubles-perturbative-triples (CCSD(T)) calcns. are presented for atoms and mols. with a focus on the dependence of correlation and binding energies on the magnetic field. (c) 2015 American Institute of Physics.**10**Hampe, F.; Stopkowicz, S. Equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fields.*J. Chem. Phys.*2017,*146*, 154105, DOI: 10.1063/1.4979624Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXmtlKhsLk%253D&md5=15e2462d7406ee3e6fa60a14f3b2c9e2Equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fieldsHampe, Florian; Stopkowicz, StellaJournal of Chemical Physics (2017), 146 (15), 154105/1-154105/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A program for the direct calcn. of excitation energies of atoms and mols. in strong magnetic fields is presented. The implementation includes the equation-of-motion coupled-cluster singles-doubles (EOM-CCSD) method for electronically excited states as well as its spin-flip variant. Differences to regular EOM-CCSD implementations are due to the appearance of the canonical angular-momentum operator in the Hamiltonian causing the wave function to become complex. The gauge-origin problem is treated by the use of gauge-including AOs. Therefore, a modified Davidson method for diagonalizing complex non-Hermitian matrixes is used. Excitation energies for selected atoms and mols. that are of importance in the astrochem. context are presented and their dependence on the magnetic field is discussed. (c) 2017 American Institute of Physics.**11**Hampe, F.; Gross, N.; Stopkowicz, S. Full triples contribution in coupled-cluster and equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fields.*Phys. Chem. Chem. Phys.*2020,*22*, 23522– 23529, DOI: 10.1039/D0CP04169FGoogle Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXitFGqu7vO&md5=792d3a16d609d1d13c9886adc6292909Full triples contribution in coupled-cluster and equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fieldsHampe, Florian; Gross, Niklas; Stopkowicz, StellaPhysical Chemistry Chemical Physics (2020), 22 (41), 23522-23529CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Coupled-cluster as well as equation-of-motion coupled-cluster methods play an important role whenever high accuracy is warranted. Concerning excitation energies, consideration of triple excitations is typically required to reach an accuracy better than 0.1-0.3 eV. In the context of strong magnetic fields such accuracy is needed for the prediction of spectra of strongly magnetized White Dwarfs. In addn. it turns out that in order to correctly model the behavior of energies with respect to the magnetic field strength, triple excitations are required. Due to avoided crossings which are extremely often encountered in the context of strong magnetic fields, double-excitation character can be transferred between electronic states of the same symmetry. We report an implementation of the full finite-field coupled-cluster with single, double, and triple substitutions (CCSDT) and the equation-of-motion-CCSDT models and apply them to the prediction of field-dependent transition wavelengths for sodium as well as to the four lowest singlet states of the CH+ mol. in a strong magnetic field.**12**Furness, J. W.; Verbeke, J.; Tellgren, E. I.; Stopkowicz, S.; Ekström, U.; Helgaker, T.; Teale, A. M. Current Density Functional Theory Using Meta-Generalized Gradient Exchange-Correlation Functionals.*J. Chem. Theory Comput.*2015,*11*, 4169– 4181, DOI: 10.1021/acs.jctc.5b00535Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXht1OqsLbJ&md5=bd9f63dc323d39de7ddeccd9a85fe7bbCurrent Density Functional Theory Using Meta-Generalized Gradient Exchange-Correlation FunctionalsFurness, James W.; Verbeke, Joachim; Tellgren, Erik I.; Stopkowicz, Stella; Ekstrom, Ulf; Helgaker, Trygve; Teale, Andrew M.Journal of Chemical Theory and Computation (2015), 11 (9), 4169-4181CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present the self-consistent implementation of current-dependent (hybrid) meta-generalized gradient approxn. (mGGA) d. functionals using London AOs. A previously proposed generalized kinetic energy d. is utilized to implement mGGAs in the framework of Kohn-Sham c.d. functional theory (KS-CDFT). A unique feature of the nonperturbative implementation of these functionals is the ability to seamlessly explore a wide range of magnetic fields up to 1 au (∼235 kT) in strength. CDFT functionals based on the TPSS and B98 forms are investigated, and their performance is assessed by comparison with accurate coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) data. In the weak field regime, magnetic properties such as magnetizabilities and NMR shielding consts. show modest but systematic improvements over generalized gradient approxns. (GGA). However, in the strong field regime, the mGGA-based forms lead to a significantly improved description of the recently proposed perpendicular paramagnetic bonding mechanism, comparing well with CCSD(T) data. In contrast to functionals based on the vorticity, these forms are found to be numerically stable, and their accuracy at high field suggests that the extension of mGGAs to CDFT via the generalized kinetic energy d. should provide a useful starting point for further development of CDFT approxns.**13**Reimann, S.; Borgoo, A.; Austad, J.; Tellgren, E. I.; Teale, A. M.; Helgaker, T.; Stopkowicz, S. Kohn–Sham energy decomposition for molecules in a magnetic field.*Mol. Phys.*2019,*117*, 97– 109, DOI: 10.1080/00268976.2018.1495849Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhtlGhsrjL&md5=daf402b362574e5f792c3ac5acd3e0e6Kohn-Sham energy decomposition for molecules in a magnetic fieldReimann, Sarah; Borgoo, Alex; Austad, Jon; Tellgren, Erik I.; Teale, Andrew M.; Helgaker, Trygve; Stopkowicz, StellaMolecular Physics (2019), 117 (1), 97-109CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)We study the total mol. electronic energy and its Kohn-Sham components within the framework of magnetic-field d.-functional theory (BDFT), an alternative to current-dependent d.-functional theory (CDFT) for mols. in the presence of magnetic fields. For a selection of closed-shell dia- and paramagnetic mols., we investigate the dependence of the total electronic energy and its Kohn-Sham components on the magnetic field. Results obtained from commonly used d.-functional approxns. are compared with those obtained from Lieb optimisations based on magnetic-field dependent relaxed coupled-cluster singles-and-doubles (CCSD) and second-order Moller-Plesset (MP2) densities. We show that popular approx. exchange-correlation functionals at the generalised-gradient-approxn. (GGA), meta-GGA, and hybrid levels of theory provide a good qual. description of the electronic energy and its Kohn-Sham components in a magnetic field-in particular, for the diamagnetic mols. The performance of Hartree-Fock theory, MP2 theory, CCSD theory and BDFT with different exchange-correlation functionals is compared with coupled-cluster singles-doubles-perturbative-triples (CCSD(T)) theory for the perpendicular component of the magnetisability. Generalisations of the TPSS meta-GGA functional to systems in a magnetic field work well-the cTPSS functional, in particular, with a current-cor. kinetic-energy d., performs excellently, providing an accurate and balanced treatment of dia- and paramagnetic systems and outperforming MP2 theory.**14**Lehtola, S.; Dimitrova, M.; Sundholm, D. Fully numerical electronic structure calculations on diatomic molecules in weak to strong magnetic fields.*Mol. Phys.*2020,*118*, e1597989, DOI: 10.1080/00268976.2019.1597989Google ScholarThere is no corresponding record for this reference.**15**Irons, T. J. P.; Zemen, J.; Teale, A. M. Efficient Calculation of Molecular Integrals over London Atomic Orbitals.*J. Chem. Theory Comput.*2017,*13*, 3636– 3649, DOI: 10.1021/acs.jctc.7b00540Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtFChsb7L&md5=dbec737cdb32d62c96bf2c20f35ef389Efficient Calculation of Molecular Integrals over London Atomic OrbitalsIrons, Tom J. P.; Zemen, Jan; Teale, Andrew M.Journal of Chemical Theory and Computation (2017), 13 (8), 3636-3649CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The use of London AOs (LAOs) in a nonperturbative manner enables the detn. of gauge-origin invariant energies and properties for mol. species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calcns. for mol. systems is the evaluation of mol. integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiencies of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP), and Rys quadrature schemes are compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and wts., enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalized algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta; thus, a simple mixed scheme is put forward that selects the most efficient approach to calc. the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm.**16**David, G.; Irons, T. J. P.; Fouda, A. E. A.; Furness, J. W.; Teale, A. M. SCF Methods for Excited States in Strong Magnetic Fields. Manuscript in preparation, 2021.Google ScholarThere is no corresponding record for this reference.**17**Wibowo, M.; Irons, T. J. P.; Teale, A. M. Modelling ultrafast electron dynamics in strong magnetic fields using real-time time-dependent electronic structure methods.*J. Chem. Theory Comput.*2021, DOI: 10.1021/acs.jctc.0c01269 .Google ScholarThere is no corresponding record for this reference.**18**Angel, J. R. P. Magnetism in white dwarfs.*Astrophys. J.*1977,*216*, 1, DOI: 10.1086/155436Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2sXltV2ht7w%253D&md5=738df5f09f6429a0a61fe1af083fbf67Magnetism in white dwarfsAngel, J. R. P.Astrophysical Journal (1977), 216 (1, Pt. 1), 1-17CODEN: ASJOAB; ISSN:0004-637X.A few percent of all white dwarfs are strongly magnetic. Ten examples are now known, of which half have spectra which show Zeeman splitting in lines of H, He, or CH in magnetic fields of 5 to 25 × 106 gauss. Two of these have sharply defined Zeeman subcomponents, indicative of very uniform surface fields. The remaining 5 have still stronger fields, such that the spectral features if present are weak and of uncertain origin. In these objects the magnetic field is identified by the elliptical polarization of the optical continuum, and is of order 108 gauss. Within the small sample of 10 there is some evidence that the magnetic field modifies the normal extreme atm. compns. of white dwarfs. Most of the magnetic white dwarfs show no spectral or polarization variations, and may be rotating very slowly (P > 10 years). However, 2 are identified as oblique rotators with periods of the order of hours, in line with ests. of nonmagnetic white dwarfs. Other measurable properties of magnetic white dwarfs do not seem remarkably different from white dwarfs in general. The fact tha they are not esp. hot means that the time scale for field decay is comparable to or longer than that for cooling.**19**Lai, D. Matter in strong magnetic fields.*Rev. Mod. Phys.*2001,*73*, 629– 662, DOI: 10.1103/RevModPhys.73.629Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXptVyht7Y%253D&md5=47712d6d4aa8fd63065cc9b5ee72e123Matter in strong magnetic fieldsLai, DongReviews of Modern Physics (2001), 73 (3), 629-661CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. The properties of matter are drastically modified by strong magnetic fields, B»me2e3c/ℏ3 = 2.35 × 109 G (1 G = 10-4T), as are typically found on the surfaces of neutron stars. In such strong magnetic fields, the Coulomb force on an electron acts as a small perturbation compared to the magnetic force. The strong-field condition can also be mimicked in lab. semiconductors. Because of the strong magnetic confinement of electrons perpendicular to the field, atoms attain a much greater binding energy compared to the zero-field case, and various other bound states become possible, including mol. chains and 3-dimensional condensed matter. This article reviews the electronic structure of atoms, mols., and bulk matter, as well as the thermodn. properties of dense plasma, in strong magnetic fields, 109 G«B1016 G. The focus is on the basic phys. pictures and approx. scaling relations, although various theor. approaches and numerical results are also discussed. For a neutron star surface composed of light elements such as H or He, the outermost layer constitutes a nondegenerate, partially ionized Coulomb plasma if B1015 G (at temp. T 106 K), and may be as a condensed liq. if the magnetic field is stronger (and T106 K). For an Fe surface, the outermost layer of the neutron star can be in a gaseous or a condensed phase, depending on the cohesive property of the Fe condensate.**20**Ferrario, L.; de Martino, D.; Gänsicke, B. T. Magnetic White Dwarfs.*Space Sci. Rev.*2015,*191*, 111– 169, DOI: 10.1007/s11214-015-0152-0Google ScholarThere is no corresponding record for this reference.**21**Xu, S.; Jura, M.; Koester, D.; Klein, B.; Zuckerman, B. Discovery of molecular hydrogen in white dwarf atmospheres.*Astrophys. J., Lett.*2013,*766*, L18, DOI: 10.1088/2041-8205/766/2/L18Google Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXht1ajtbrP&md5=2e94bb4aade7867066121741fc8a936aDiscovery of molecular hydrogen in white dwarf atmospheresXu, S.; Jura, M.; Koester, D.; Klein, B.; Zuckerman, B.Astrophysical Journal, Letters (2013), 766 (2), L18/1-L18/3, 3 pp.CODEN: AJLEEY; ISSN:2041-8213. (IOP Publishing Ltd.)With the Cosmic Origins Spectrograph on board the Hubble Space Telescope, we have detected mol. hydrogen in the atmospheres of three white dwarfs with effective temps. below 14,000 K, G29-38, GD 133 and GD 31. This discovery provides new independent constraints on the stellar temp. and surface gravity of white dwarfs.**22**Compernolle, S.; Chibotaru, L. F.; Ceulemans, A. Vortices and Their Relation to Ring Currents and Magnetic Moments in Nanographenes in High Magnetic Field.*J. Phys. Chem. B*2006,*110*, 19340– 19351, DOI: 10.1021/jp063947hGoogle Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XptVyltrY%253D&md5=ad9a565162fd37830c9217b59311650aVortices and Their Relation to Ring Currents and Magnetic Moments in Nanographenes in High Magnetic FieldCompernolle, S.; Chibotaru, L. F.; Ceulemans, A.Journal of Physical Chemistry B (2006), 110 (39), 19340-19351CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)Much attention has been paid to the role of vortices in the magnetic response properties of superconductors, but less so for mol. systems. Here we present a theor. anal. on nanographenes subject to a strong homogeneous magnetic field. The anal. is based on the simple H.ovrddot.uckel-London model, for which we derive the topol. definition of vorticity. The results are confirmed by a more elaborate model that includes nonnearest neighbor interaction, the explicit presence of nuclei and all terms due to the magnetic field. We find that due to frontier orbital intersections, large changes in magnetic dipole moments occur. Orbital energy min. and maxima can be related to change of vortex patterns with flux.**23**Murdin, B.; Li, J.; Pang, M.; Bowyer, E.; Litvinenko, K.; Clowes, S.; Engelkamp, H.; Pidgeon, C.; Galbraith, I.; Abrosimov, N.; Riemann, H.; Pavlov, S.; Hübers, H.-W.; Murdin, P. Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf stars.*Nat. Commun.*2013,*4*, 1469, DOI: 10.1038/ncomms2466Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC3sznslKhtg%253D%253D&md5=b423aca7f3dee202fb9e1c4ec327a9c2Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf starsMurdin B N; Li Juerong; Pang M L Y; Bowyer E T; Litvinenko K L; Clowes S K; Engelkamp H; Pidgeon C R; Galbraith I; Abrosimov N V; Riemann H; Pavlov S G; Hubers H-W; Murdin P GNature communications (2013), 4 (), 1469 ISSN:.Laboratory spectroscopy of atomic hydrogen in a magnetic flux density of 10(5) T (1 gigagauss), the maximum observed on high-field magnetic white dwarfs, is impossible because practically available fields are about a thousand times less. In this regime, the cyclotron and binding energies become equal. Here we demonstrate Lyman series spectra for phosphorus impurities in silicon up to the equivalent field, which is scaled to 32.8 T by the effective mass and dielectric constant. The spectra reproduce the high-field theory for free hydrogen, with quadratic Zeeman splitting and strong mixing of spherical harmonics. They show the way for experiments on He and H(2) analogues, and for investigation of He(2), a bound molecule predicted under extreme field conditions.**24***LONDON*, A quantum chemistry program for plane-wave/GTO hybrid basis sets and finite magnetic field calculations; http://londonprogram.org (accessed December 16, 2020).Google ScholarThere is no corresponding record for this reference.**25***BAGEL, Brilliantly Advanced General Electronic-Structure Library*; http://nubakery.org (accessed December 16, 2020).Google ScholarThere is no corresponding record for this reference.**26**Williams-Young, D. B.; Petrone, A.; Sun, S.; Stetina, T. F.; Lestrange, P.; Hoyer, C. E.; Nascimento, D. R.; Koulias, L.; Wildman, A.; Kasper, J.; Goings, J. J.; Ding, F.; DePrince, A. E.; Valeev, E. F.; Li, X. The Chronus Quantum software package.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2020,*10*, e1436, DOI: 10.1002/wcms.1436Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXmvV2qsrw%253D&md5=7c0938366c14e76b901d9a54a3f4f497The Chronus Quantum software packageWilliams-Young, David B.; Petrone, Alessio; Sun, Shichao; Stetina, Torin F.; Lestrange, Patrick; Hoyer, Chad E.; Nascimento, Daniel R.; Koulias, Lauren; Wildman, Andrew; Kasper, Joseph; Goings, Joshua J.; Ding, Feizhi; DePrince, A. Eugene, III; Valeev, Edward F.; Li, XiaosongWiley Interdisciplinary Reviews: Computational Molecular Science (2020), 10 (2), e1436CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)The Chronus Quantum (ChronusQ) software package is an open source (under the GNU General Public License v2) software infrastructure which targets the soln. of challenging problems that arise in ab initio electronic structure theory. Special emphasis is placed on the consistent treatment of time dependence and spin in the electronic wave function, as well as the inclusion of relativistic effects in said treatments. In addn., ChronusQ provides support for the inclusion of uniform finite magnetic fields as external perturbations through the use of gauge-including AOs. ChronusQ is a parallel electronic structure code written in modern C++ which utilizes both message passing implementation and shared memory (OpenMP) parallelism. In addn. to the examn. of the current state of code base itself, a discussion regarding ongoing developments and developer contributions will also be provided. This article is categorized under:Software > Quantum Chem. Electronic Structure Theory > Ab Initio Electronic Structure Methods Electronic Structure Theory > D. Functional Theory.**27**Pausch, A.; Klopper, W. Efficient evaluation of three-centre two-electron integrals over London orbitals.*Mol. Phys.*2020,*118*, e1736675, DOI: 10.1080/00268976.2020.1736675Google Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXktl2gtL4%253D&md5=ec2a6c383ef9fbb479822f7406b61463Efficient evaluation of three-centre two-electron integrals over London orbitalsPausch, Ansgar; Klopper, WimMolecular Physics (2020), 118 (21-22), e1736675/1-e1736675/11CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)A review. The nonperturbative calcn. of mol. properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London AOs (LAOs). With these orbitals, the calcn. of four-center electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resoln.-of-the-identity (RI) approxn. for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approxn. is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-center integrals in the RI approxn. remain real-valued, and the three-center integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calcn. of four-center ERIs over LAOs, using the RI approxn. thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an addnl. factor of at least two. By using other well-established methods such as Cauchy-Schwarz screening, the difference-d. approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calcns. in magnetic fields can be increased even further.**28***QUEST, A rapid development platform for Quantum Electronic Structure Techniques*, 2017; quest.codes (accessed December 16, 2020).Google ScholarThere is no corresponding record for this reference.**29**London, F. Théorie quantique des courants interatomiques dans les combinaisons aromatiques.*J. Phys. Radium*1937,*8*, 397– 409, DOI: 10.1051/jphysrad:01937008010039700Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaA1cXitVahsA%253D%253D&md5=3539386fad997734786f7ad49466deaeQuantum theory of interatomic currents in aromatic compoundsLondon, F.Journal de Physique et le Radium (1937), 8 (), 397-409CODEN: JPRAAJ; ISSN:0368-3842.Math. study of the anomalous anisotropic diamagnetism observed in aromatic compds. This is explained on the basis of interat. elec. currents peculiar to these compds. Benzene, naphthalene, anthracene, biphenyl, pyrene and phenanthrene are treated as examples.**30**Reynolds, R. D.; Shiozaki, T. Fully relativistic self-consistent field under a magnetic field.*Phys. Chem. Chem. Phys.*2015,*17*, 14280– 14283, DOI: 10.1039/C4CP04027AGoogle Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhslags7bJ&md5=3e856c62040ee1ae6065b4b9f621f3e2Fully relativistic self-consistent field under a magnetic fieldReynolds, Ryan D.; Shiozaki, ToruPhysical Chemistry Chemical Physics (2015), 17 (22), 14280-14283CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We present a gauge-invariant implementation of the four-component Dirac-Hartree-Fock method for simulating the electronic structure of heavy element complexes in magnetic fields. The addnl. cost assocd. with the magnetic field is shown to be only 10-13% of that at zero field. The Dirac-Hartree-Fock wave function is constructed from gauge-including AOs. The so-called restricted magnetic balance is used to generate 2-spinor basis functions for the small component. The mol. integrals for the Coulomb and Gaunt interactions are computed using d. fitting. Our efficient, parallel implementation allows for simulating the electronic structure of mols. contg. more than 100 atoms with a few heavy elements under magnetic fields.**31**Schlegel, H. B.; Frisch, M. J. Transformation between Cartesian and pure spherical harmonic Gaussians.*Int. J. Quantum Chem.*1995,*54*, 83– 87, DOI: 10.1002/qua.560540202Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXktlOmtLY%253D&md5=2971b5414f3b3b740f33a15b756323abTransformation between Cartesian and pure spherical harmonic GaussiansSchlegel, H. Bernhard; Freisch, Michael J.International Journal of Quantum Chemistry (1995), 54 (2), 83-7CODEN: IJQCB2; ISSN:0020-7608. (Wiley)Spherical Gaussians can be expressed as linear combinations of the appropriate Cartesian Gaussians. General expressions for the transformation coeffs. are given. Values for the transformation coeffs. are tabulated up to h-type functions.**32**Helgaker, T.; Jørgensen, P.; Olsen, J.*Molecular Electronic-Structure Theory*; John Wiley & Sons, 2000; DOI: 10.1002/9781119019572 .Google ScholarThere is no corresponding record for this reference.**33**McMurchie, L.; Davidson, E. One- and two-electron integrals over Cartesian Gaussian functions.*J. Comput. Phys.*1978,*26*, 218– 231, DOI: 10.1016/0021-9991(78)90092-XGoogle Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE1cXhsFyksb4%253D&md5=119248f25c59392ff1d9ce1c7285b3c7One- and two-electron integrals over Cartesian Gaussian functionsMcMurchie, Larry E.; Davidson, Ernest R.Journal of Computational Physics (1978), 26 (2), 218-31CODEN: JCTPAH; ISSN:0021-9991.A formalism was developed that allows overlap, kinetic-energy, potential-energy, and electron-repulsion integrals over Cartesian Gaussian wave functions to be expressed in very compact forms involving easily calcd. auxiliary functions. Similar formulas involving the same auxiliary functions are given for the common charge moments, elec.-field operators, and spin-interaction operators. Recursion relations are given for the auxiliary functions, which make it possible to use Gaussian wave functions having arbitrarily large angular momentum. An algorithm is given for calcg. electron-repulsion integrals.**34**McMurchie, L. E.; Davidson, E. R. One- and two-electron integrals over Cartesian Gaussian functions.*J. Comput. Phys.*1978,*26*, 218– 231, DOI: 10.1016/0021-9991(78)90092-XGoogle Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE1cXhsFyksb4%253D&md5=119248f25c59392ff1d9ce1c7285b3c7One- and two-electron integrals over Cartesian Gaussian functionsMcMurchie, Larry E.; Davidson, Ernest R.Journal of Computational Physics (1978), 26 (2), 218-31CODEN: JCTPAH; ISSN:0021-9991.A formalism was developed that allows overlap, kinetic-energy, potential-energy, and electron-repulsion integrals over Cartesian Gaussian wave functions to be expressed in very compact forms involving easily calcd. auxiliary functions. Similar formulas involving the same auxiliary functions are given for the common charge moments, elec.-field operators, and spin-interaction operators. Recursion relations are given for the auxiliary functions, which make it possible to use Gaussian wave functions having arbitrarily large angular momentum. An algorithm is given for calcg. electron-repulsion integrals.**35**Fortunelli, A.; Salvetti, O. Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functions.*Int. J. Quantum Chem.*1993,*48*, 257– 265, DOI: 10.1002/qua.560480407Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXhsFejsA%253D%253D&md5=32a6b01291a224985215ef9ec0097335Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functionsFortunelli, Alessandro; Salvetti, OrianoInternational Journal of Quantum Chemistry (1993), 48 (4), 257-65CODEN: IJQCB2; ISSN:0020-7608.Recurrence relations are derived for the evaluation of two-electron repulsion integrals (ERIs) over Hermite and spherical Gaussian functions. Through such relations, a generic ERI or ERI deriv. may be reduced to "basic" integrals, i.e., true and auxiliary integrals involving only zero angular momentum functions.**36**Reine, S.; Tellgren, E.; Helgaker, T. A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians.*Phys. Chem. Chem. Phys.*2007,*9*, 4771, DOI: 10.1039/b705594cGoogle Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXptlSqsrc%253D&md5=04c2f3ae05a68ce9d39cc8909765c3d8A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic GaussiansReine, Simen; Tellgren, Erik; Helgaker, TrygvePhysical Chemistry Chemical Physics (2007), 9 (34), 4771-4779CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Utilizing the fact that solid-harmonic combinations of Cartesian and Hermite Gaussian AOs are identical, a new scheme for the evaluation of mol. integrals over solid-harmonic AOs is presented, where the integration is carried out over Hermite rather than Cartesian AOs. Since Hermite Gaussians are defined as derivs. of spherical Gaussians, the corresponding mol. integrals become the derivs. of integrals over spherical Gaussians, whose transformation to the solid-harmonic basis is performed in the same manner as for integrals over Cartesian Gaussians, using the same expansion coeffs. The presented solid-harmonic Hermite scheme simplifies the evaluation of deriv. mol. integrals, since differentiation by nuclear coordinates merely increments the Hermite quantum nos., thereby providing a unified scheme for undifferentiated and differentiated four-center mol. integrals. For two- and three-center two-electron integrals, the solid-harmonic Hermite scheme is particularly efficient, significantly reducing the cost relative to the Cartesian scheme.**37**Colle, R.; Fortunelli, A.; Simonucci, S. A mixed basis set of plane waves and Hermite-Gaussian functions. Analytic expressions of prototype integrals.*Nuovo Cimento Soc. Ital. Fis., D*1987,*9*, 969– 977, DOI: 10.1007/BF02464849Google ScholarThere is no corresponding record for this reference.**38**Colle, R.; Fortunelli, A.; Simonucci, S. Hermite-Gaussian functions modulated by plane waves: a general basis set for bound and continuum states.*Nuovo Cimento Soc. Ital. Fis., D*1988,*10*, 805– 818, DOI: 10.1007/BF02450141Google ScholarThere is no corresponding record for this reference.**39**Tachikawa, M.; Shiga, M. Evaluation of atomic integrals for hybrid Gaussian type and plane-wave basis functions via the McMurchie-Davidson recursion formula.*Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top.*2001,*64*, 056706, DOI: 10.1103/PhysRevE.64.056706Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXotVKgur0%253D&md5=4765ba25da04adee7437836b6d14b0dfEvaluation of atomic integrals for hybrid Gaussian type and plane-wave basis functions via the McMurchie-Davidson recursion formulaTachikawa, Masanori; Shiga, MotoyukiPhysical Review E: Statistical, Nonlinear, and Soft Matter Physics (2001), 64 (5-2), 056706/1-056706/4CODEN: PRESCM ISSN:. (American Physical Society)A convenient formalism is developed for the evaluation of at. integrals composed of a hybrid Gaussian type function and plane-wave (GTF-PW) basis set, based upon the recursion scheme proposed by McMurchie and Davidson [L. E. McMurchie and E. R. Davidson, J. Comput. Phys. 26, 218 (1978)] which was originally for Gaussian type basis functions. We show that revisions of recursion relations in the original article are necessary in order to allow systematic prodn. of overlap, kinetic energy, nuclear attraction, and electron repulsion integrals in compact forms. Involving easy calcn. of complex incomplete gamma functions, the recursion relations enable the use of hybrid GTF-PW basis functions with arbitrarily large angular momentum. This basis function can be applied to the first-principle calcn. for solids involving localized electron orbitals.**40**Kanno, M.; Kato, T.; Kono, H.; Fujimura, Y.; Faisal, F. H. M. Incorporation of a wave-packet propagation method into the S-matrix framework: Investigation of the effects of excited state dynamics on intense-field ionization.*Phys. Rev. A: At., Mol., Opt. Phys.*2005,*72*, 033418, DOI: 10.1103/PhysRevA.72.033418Google Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXhtFCmsbnF&md5=dbefcfb046e6d470fd48147f59bf3944Incorporation of a wave-packet propagation method into the S-matrix framework: Investigation of the effects of excited state dynamics on intense-field ionizationKanno, Manabu; Kato, Tsuyoshi; Kono, Hirohiko; Fujimura, Yuichi; Faisal, Farhad H. M.Physical Review A: Atomic, Molecular, and Optical Physics (2005), 72 (3, Pt. B), 033418/1-033418/14CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The authors propose a theor. method for study of ionization of atoms and mols. in intense laser fields that copes with the effects of excited state dynamics (or intramol. electronic dynamics). The time-evolving wave packet composed of only bound electronic states, |Φi(t)〉, is introduced into a framework of the intense-field S-matrix theory. Then, the effects of both Coulomb field and radiation field on the bound electron(s) are well described by |Φi(t)〉, while the effects of a radiation field on a freed electron are also treated in a nonperturbative way. The authors have applied the theory to ionization of H and H2+ in ultrashort intense laser pulses. Although only a small no. of Gaussian functions were used in the expansion of |Φi(t)〉, the present method can quant. reproduce the features of enhanced ionization of H2+ obtained by an accurate grid propagation method. This agreement supports the view that field-induced population transfer between the lowest two electronic states triggers the enhancement of ionization at large internuclear distances. The authors also applied the method to calc. the photoelectron momentum distribution of H in an intense near-IR field. A broad low intensity component due to rescattering appears in the distribution of the momentum perpendicular to the polarization direction of an applied laser field, as obsd. in the expts. of single ionization of noble gas atoms. The present method provides a practical way of properly describing the nonperturbative nature of field-induced dynamics of an electron (or electrons) in the presence of both Coulomb and radiation fields.**41**Obara, S.; Saika, A. Efficient recursive computation of molecular integrals over Cartesian Gaussian functions.*J. Chem. Phys.*1986,*84*, 3963, DOI: 10.1063/1.450106Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL28XitFeguro%253D&md5=7643c24c26bd4ea9e37424fb8cf66935Efficient recursive computation of molecular integrals over Cartesian Gaussian functionsObara, S.; Saika, A.Journal of Chemical Physics (1986), 84 (7), 3963-74CODEN: JCPSA6; ISSN:0021-9606.Recurrence expressions for calcg. various types of mol. integrals over Cartesian Gaussian functions were derived by using the recurrence formula for three-center overlap integrals. A no. of characteristics inherent in the recursive formalism allowed an efficient algorithm to be developed for mol.-integral computations. With respect to electron-repulsion integrals and their derivs., the present algorithm, with a significant saving of computer time, was superior to other currently available methods. A long innermost loop incorporated in the present scheme facilitates a fast computation on a vector-processing computer.**42**Obara, S.; Saika, A. General recurrence formulas for molecular integrals over Cartesian Gaussian functions.*J. Chem. Phys.*1988,*89*, 1540, DOI: 10.1063/1.455717Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXltlGitbc%253D&md5=7598cac04c1b8129a3084cb66b237930General recurrence formulas for molecular integrals over Cartesian Gaussian functionsObara, S.; Saika, A.Journal of Chemical Physics (1988), 89 (3), 1540-59CODEN: JCPSA6; ISSN:0021-9606.General recurrence formulas for various types of one- and two-electron mol. integrals over Cartesian Gaussian functions are derived by introducing basic integrals. These formulas are capable of dealing with (1) mol. integrals with any spatial operators in the nonrelativistic forms of the relativistic wave equations; (2) those with the kernel of the Fourier transform; (3) those with arbitrarily defined spatial operators so far as the integrals can be expressed in terms of the basic integrals; and (4) any order of their derivs. with respect to the function centers in the above integrals. Thus, the present formulation can cover a large class of mol. integrals necessary for theor. studies of mol. systems by ab initio calcns., and furthermore provides us with an efficient scheme of computing them by virtue of its recursive nature.**43**Shavitt, I.*Methods in Computational Physics*; Academic Press: New York, 1963; Vol. 3; pp 1– 45.Google ScholarThere is no corresponding record for this reference.**44**Saunders, V. R.*Computational Techniques in Quantum Chemistry and Molecular Physics*; Springer: Dordrecht, The Netherlands, 1975; pp 347– 424.Google ScholarThere is no corresponding record for this reference.**45**Gill, P. M. W.*Adv. Quantum Chem.*; Elsevier: BV, 1994; pp 141– 205.Google ScholarThere is no corresponding record for this reference.**46**Boys, S. F. Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System.*Proc. R. Soc. A*1950,*200*, 542– 554, DOI: 10.1098/rspa.1950.0036Google Scholar46https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaG3cXktVShtA%253D%253D&md5=1bf132315d97130502916898345e9b31Electronic wave functions. I. A general method of calculation for the stationary states of any molecular systemBoys, S. F.Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1950), 200 (), 542-54CODEN: PRLAAZ; ISSN:1364-5021.By taking Gaussian functions, and functions derived from these by differentiation with respect to the parameters, complete systems of functions can be constructed appropriate to any mol. problem, and all the necessary integrals can be explicitly evaluated. The only obstacle to the evaluation of wave functions of any required degree of accuracy is the labor of computation. The methods developed give for the first time a quant. method of evaluating the stationary-state wave functions and energy levels of all atoms and mols. to any required degree of accuracy.**47**Čársky, P.; Polášek, M. Incomplete Gamma Fm(x) Functions for Real Negative and Complex Arguments.*J. Comput. Phys.*1998,*143*, 259– 265, DOI: 10.1006/jcph.1998.5975Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXkt1Oiu7o%253D&md5=7a63cbbd4b75f93901ec8b20450c8f01Incomplete gamma Fm(x) functions for real negative and complex argumentsCarsky, Petr; Polasek, MartinJournal of Computational Physics (1998), 143 (1), 259-265CODEN: JCTPAH; ISSN:0021-9991. (Academic Press)Incomplete gamma functions Fm(x), originally defined and used in the electronic structure theory, have been examd. from the viewpoint of electron-mol. scattering theory for their possible use in calcn. of two-electron integrals in a mixed Gaussian and plane-wave basis set. Effective calcn. of Fm(z) functions with a complex argument is discussed. (c) 1998 Academic Press.**48**Ishida, K. Accurate and fast algorithm of the molecular incomplete gamma function with a complex argument.*J. Comput. Chem.*2004,*25*, 739– 748, DOI: 10.1002/jcc.20002Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXis1Kktb4%253D&md5=baa4c691923a9780dc754aad86097e1aAccurate and fast algorithm of the molecular incomplete gamma function with a complex argumentIshida, KazuhiroJournal of Computational Chemistry (2004), 25 (5), 739-748CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)Several efficient algorithms for the accurate and fast calcn. of the mol. incomplete gamma function Fm(z) with a complex argument z are developed. The complex incomplete gamma function is arising in mol. integrals over the gauge-including AOs. Two kinds of algorithms are recommended: (1) a high-precision version and (2) a fast version. The high-precision version is able to guarantee 15 significant figures (10-15 in the relative error) and the fast version is able to guarantee 12 significant figures (10-12 in the relative error), at worst, within the double-precision arithmetic. The fast version is about 5-20 times faster than the high-precision version. For most mol. calcns., the fast version will give a satisfied precision.**49**Mathar, R. J. Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument.*Numer. Algorithms*2004,*36*, 247– 264, DOI: 10.1023/B:NUMA.0000040063.91709.58Google ScholarThere is no corresponding record for this reference.**50**Helgaker, T.; Taylor, P. R. On the evaluation of derivatives of Gaussian integrals.*Theor. Chim. Acta*1992,*83*, 177– 183, DOI: 10.1007/BF01132826Google ScholarThere is no corresponding record for this reference.**51**King, H. F.; Dupuis, M. Numerical integration using Rys polynomials.*J. Comput. Phys.*1976,*21*, 144– 165, DOI: 10.1016/0021-9991(76)90008-5Google ScholarThere is no corresponding record for this reference.**52**Dupuis, M.; Rys, J.; King, H. F. Evaluation of molecular integrals over Gaussian basis functions.*J. Chem. Phys.*1976,*65*, 111– 116, DOI: 10.1063/1.432807Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE28XkvF2nsb4%253D&md5=1ede1bceba64efb430a3a383443c55ceEvaluation of molecular integrals over Gaussian basis functionsDupuis, Michel; Rys, John; King, Harry F.Journal of Chemical Physics (1976), 65 (1), 111-16CODEN: JCPSA6; ISSN:0021-9606.The efficient computation of the ubiquitous electron repulsion integral in mol. quantum mechanics was studied. Differences and similarities in organization of existing Gaussian integral programs are discussed, and a new strategy is developed. An anal. based on the theory of orthogonal polynomials yields a general formula for basis functions of arbitrarily high angular momentum. This method is simple, numerically well behaved, and was incorporated into a new mol. SCF program HONDO. Preliminary tests indicate that it is competitive with existing methods esp. for highly angularly dependent functions.**53**Rys, J.; Dupuis, M.; King, H. F. Computation of electron repulsion integrals using the Rys quadrature method.*J. Comput. Chem.*1983,*4*, 154– 157, DOI: 10.1002/jcc.540040206Google Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXit1Wqu7s%253D&md5=f8d3d4f8d5c3f1fc2492554f38b88bdcComputation of electron repulsion integrals using the Rys quadrature methodRys, J.; Dupuis, M.; King, H. F.Journal of Computational Chemistry (1983), 4 (2), 154-7CODEN: JCCHDD; ISSN:0192-8651.Following an earlier proposal to evaluate electron repulsion integrals over Gaussian basis functions by a numerical quadrature based on a set of orthogonal polynomials (Rys polynomials), a computational procedure is outlined for efficient evaluation of the two-dimensional integrals. Compact recurrence formulas for the integrals make the method particularly fitted to handle high-angular-momentum basis functions.**54**Čársky, P.; Polášek, M. Evaluation of Molecular Integrals in a Mixed Gaussian and Plane-Wave Basis by Rys Quadrature.*J. Comput. Phys.*1998,*143*, 266– 277, DOI: 10.1006/jcph.1998.5976Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXkt1Oiu7s%253D&md5=68620231706ab0aae7fed92b069be57eEvaluation of molecular integrals in a mixed Gaussian and plane-wave basis by Rys quadratureCarsky, Petr; Polasek, MartinJournal of Computational Physics (1998), 143 (1), 266-277CODEN: JCTPAH; ISSN:0021-9991. (Academic Press)We report on the use of Rys numerical quadrature for the calcn. of two-electron exchange integrals contg. two Gaussians and two plane-wave functions, and two-electron integrals contg. three Gaussians and one plane-wave function. Generally, the Rys polynomials for this mixed basis set are complex. We present formulas for obtaining their roots and wts. that are also generally complex. Rys numerical quadrature provides an alternative method for calcn. of integrals of this type that are encountered in the electron-mol. scattering theory. (c) 1998 Academic Press.**55**Head-Gordon, M.; Pople, J. A. A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relations.*J. Chem. Phys.*1988,*89*, 5777, DOI: 10.1063/1.455553Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXivFajuw%253D%253D&md5=c986058fd86ea827e91bb3ea2ac57519A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relationsHead-Gordon, Martin; Pople, John A.Journal of Chemical Physics (1988), 89 (9), 5777-86CODEN: JCPSA6; ISSN:0021-9606.An efficient method is presented for evaluating two-electron Cartesian Gaussian integrals, and their first derivs. with respect to nuclear coordinates. It is based on the recurrence relation (RR) of Obara and Saika (1986), and an addnl. new RR, which are combined together in a general algorithm applicable to any angular momenta. This algorithm exploits the fact that the new RR can be applied outside contraction loops. By floating point operation counts and comparative timings it is shown to be generally superior to existing methods, particularly for basis sets contg. d functions.**56**Lindh, R.; Ryu, U.; Liu, B. The reduced multiplication scheme of the Rys quadrature and new recurrence relations for auxiliary function based two-electron integral evaluation.*J. Chem. Phys.*1991,*95*, 5889, DOI: 10.1063/1.461610Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXmtl2qtLo%253D&md5=0915bf9171b22ea14f8a64e9df982b58The reduced multiplication scheme of the Rys quadrature and new recurrence relations for auxiliary function based two-electron integral evaluationLindh, R.; Ryu, U.; Liu, B.Journal of Chemical Physics (1991), 95 (8), 5889-97CODEN: JCPSA6; ISSN:0021-9606.A reduced-multiplication algorithm for the Rys quadrature is presented. The method is based on new ways in which the Rys quadrature can be developed if it is implemented together with the transfer equation applied to the contracted integrals. In parallel to the new algorithm for the Rys quadrature, improvements are suggested to the auxiliary-function-based algorithms. The two new methods have very favorable theor. floating point operation (FLOP) counts as compared to other methods. It is noted that the only significant difference in performance of the two new methods is due to the vectorizability of the presented algorithms. In order to exhibit this, both methods were implemented in the integral program SEWARD. Timings are presented for comparisons with other implementations. Finally, it is demonstrated how the transfer equation in connection with the use of spherical harmonic Gaussians offers a very attractive path to compute the two-electron integrals of such basis functions. It is demonstrated both theor. and with actual performance that the use of spherical harmonic Gaussians offers a clear advantage over the traditional evaluation of the two-electron integrals in the Cartesian Gaussian basis.**57**Lindh, R. The reduced multiplication scheme of the Rys-Gauss quadrature for 1st order integral derivatives.*Theor. Chim. Acta*1993,*85*, 423– 440, DOI: 10.1007/BF01112982Google Scholar57https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXls1Oqur0%253D&md5=f433efff3d647af6c17351fee6d70a31The reduced multiplication scheme of the Rys-Gauss quadrature for 1st order integral derivativesLindh, RolandTheoretica Chimica Acta (1993), 85 (6), 423-40CODEN: TCHAAM; ISSN:0040-5744.An implementation of the reduced multiplication scheme of the Rys-Gauss quadrature to compute the gradients of electron repulsion integrals is discussed. The Rys-Gauss quadrature is very suitable for efficient utilization of simplifications as offered by the direct computation of symmetry adapted gradients and the use of the translational invariance of the integrals. The introduction of the so-called intermediate products is also demonstrated to further reduce the floating point operation count. Two prescreening techniques based on the 2nd order d. matrix in the basis of the uncontracted Gaussian functions is proposed and investigated in this paper. It is not necessary to employ the Cauchy-Schwarz inequality to achieve efficient prescreening. All the features mentioned above were demonstrated by their implementation into the gradient program ALASKA. The paper offers a theor. and practical assessment of the modified Rys-Gauss quadrature in comparison with other methods and implementations and a detailed anal. of the behavior of the method as suggested above as a function of changes with respect to symmetry, basis set quality, mol. size, and prescreening threshold.**58**Komornicki, A.; Ishida, K.; Morokuma, K.; Ditchfield, R.; Conrad, M. Efficient determination and characterization of transition states using ab-initio methods.*Chem. Phys. Lett.*1977,*45*, 595– 602, DOI: 10.1016/0009-2614(77)80099-7Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2sXht1Omsrw%253D&md5=9f2041a5838eb8d70a4653813bf06813Efficient determination and characterization of transition states using ab-initio methodsKomornicki, Andrew; Ishida, Kazuhiro; Morokuma, Keiji; Ditchfield, Robert; Conrad, MorganChemical Physics Letters (1977), 45 (3), 595-602CODEN: CHPLBC; ISSN:0009-2614.The gradient of the potential energy with respect to the nuclear coordinates was calcd. by using ab-initio single-determinant MO methods. The calcd. gradient was used together with very efficient minimization methods to locate and characterize transition states on many-dimensional potential-energy surfaces. Previously such methods have only been applied to semiempirical potential functions. Although the calcn. of the gradient in addn. to the energy increases the computational time by about a factor of four, the feasibility of these calcns. was demonstrated by locating the transition state for the model rearrangement of HNC to HCN by using both minimal (STO-3G) and split-valence-shell (4-31G) basis sets. Further use of such methods is discussed in the direct application of ab-initio wave functions to dynamical investigations.**59**Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas.*Phys. Rev.*1964,*136*, B864– B871, DOI: 10.1103/PhysRev.136.B864Google ScholarThere is no corresponding record for this reference.**60**Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects.*Phys. Rev.*1965,*140*, A1133– A1138, DOI: 10.1103/PhysRev.140.A1133Google ScholarThere is no corresponding record for this reference.**61**Vignale, G.; Rasolt, M. Density-functional theory in strong magnetic fields.*Phys. Rev. Lett.*1987,*59*, 2360– 2363, DOI: 10.1103/PhysRevLett.59.2360Google Scholar61https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXislSqtA%253D%253D&md5=6c0439dbbbddf191b1da1f73d69ed733Density-functional theory in strong magnetic fieldsVignale, G.; Rasolt, MarkPhysical Review Letters (1987), 59 (20), 2360-3CODEN: PRLTAO; ISSN:0031-9007.The current-d.-functional theory is formulated for systems in arbitrarily strong magnetic fields. A set of self-consistent equations comparable to the Kohn-Sham equations for ordinary d.-functional theory is derived, and proved to be gage invariant and to satisfy the continuity equation. The exchange-correlation energy functional Exc[n,jp] [n(r) is the d. and jp(r) is the "paramagnetic" c.d.] depends on the current via the combination v(r) = V × [jp(r)/n(r)]. An explicit formula for Exc is derived, which is local in v(r).**62**Vignale, G.; Rasolt, M. Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields.*Phys. Rev. B: Condens. Matter Mater. Phys.*1988,*37*, 10685– 10696, DOI: 10.1103/PhysRevB.37.10685Google Scholar62https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2sfhtFaqug%253D%253D&md5=08fd751039a075c8cc57ec29f67246ffCurrent- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fieldsVignale; RasoltPhysical review. B, Condensed matter (1988), 37 (18), 10685-10696 ISSN:0163-1829.There is no expanded citation for this reference.**63**Dobson, J. F. Interpretation of the Fermi hole curvature.*J. Chem. Phys.*1991,*94*, 4328– 4333, DOI: 10.1063/1.460619Google Scholar63https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXitFCrsLs%253D&md5=bd726fc23d10d626f433e386fbd26816Interpretation of the Fermi hole curvatureDobson, John F.Journal of Chemical Physics (1991), 94 (6), 4328-33CODEN: JCPSA6; ISSN:0021-9606.Two different interpretations are given for the Fermi-hole-curvature parameter in many-electron systems used recently to est. the size of the correlation hole, to clarify aspects of chem. shell structure and bonding, and to discuss mobility of the Fermi hole. The first, more straightforward interpretation involves the no. of "other" electrons to be found in a small neighborhood near a given electron. The notion of other electrons leads naturally to correlation functionals, which correctly vanish when only one electron is present. The second interpretation, made explicit by use of the Wigner pair distribution, involves the d. of relative kinetic energy of pairs of spin-parallel electrons at point r. Since, in a classical interpretation at least, the correlation hole in a nonuniform Coulomb system depends both on d. and relative kinetic energy of colliding pairs, one expects that both the Fermi hole curvature and the d. should be significant in constructing theories of the correlation energy of such systems.**64**Dobson, J. F. Alternative expressions for the Fermi hole curvature.*J. Chem. Phys.*1993,*98*, 8870– 8872, DOI: 10.1063/1.464444Google Scholar64https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXkvVSmt74%253D&md5=0623ae49360b47df1b9560a86baffe77Alternative expressions for the Fermi hole curvatureDobson, John F.Journal of Chemical Physics (1993), 98 (11), 8870-2CODEN: JCPSA6; ISSN:0021-9606.The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero sepn. r' = r for a pair of fermions in a many-fermion system. It has been used by a no. of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-fermion state and is not limited to the single-determinant case investigated previously.**65**Becke, A. D. Current-density dependent exchange-correlation functionals.*Can. J. Chem.*1996,*74*, 995– 997, DOI: 10.1139/v96-110Google Scholar65https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xks1yrsbY%253D&md5=4198a87b2a632236782239cbfafce319Current-density dependent exchange-correlation functionalsBecke, Axel D.Canadian Journal of Chemistry (1996), 74 (6), 995-997CODEN: CJCHAG; ISSN:0008-4042. (National Research Council of Canada)Previous models for exchange (Becke and Roussel, Phys. Rev. A:39, 3761 (1989)) and for correlation (Becke, J. Chem. Phys. 88, 1053 (1988)) are, in a simple and natural way, generalized to include explicit dependence on c.d. J. First-principles incorporation of J into exchange-correlation d. functionals, as proposed here, is crucial for further progress in the study of magnetic effects in d.-functional theory.**66**Becke, A. D. Current density in exchange-correlation functionals: Application to atomic states.*J. Chem. Phys.*2002,*117*, 6935– 6938, DOI: 10.1063/1.1503772Google Scholar66https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XnsF2itro%253D&md5=37fd49dbccbeadc7d61935a97183f1cfCurrent density in exchange-correlation functionals: Application to atomic statesBecke, Axel D.Journal of Chemical Physics (2002), 117 (15), 6935-6938CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An old and yet unsolved problem in d.-functional theory is the strong dependence of degenerate open-shell at. energies on the occupancy of the AOs. This arises from the fact that degenerate AOs of different ml do not have equiv. densities. Approx. d. functionals therefore give energies depending strongly on which orbitals are occupied. This problem is solved in the present work by incorporating c.d. into the calcns. using a current-d. dependent functional previously published by the author.**67**Neumann, R.; Nobes, R. H.; Handy, N. C. Exchange functionals and potentials.*Mol. Phys.*1996,*87*, 1– 36, DOI: 10.1080/00268979600100011Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XhvFegs7c%253D&md5=14030dab7e55c7c6faaf9737dbcbcb37Exchange functionals and potentialsNeumann, Ralf; Nobes, Ross H.; Handy, Nicholas C.Molecular Physics (1996), 87 (1), 1-36CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis)The commonly used exchange-correlation functionals of d. functional theory and their potentials are examd. numerically following the first such investigation of Perdew. They are also investigated for Ne and Kr. Their behavior for large gradients of the d. and for large distances is not satisfactory. In particular, the correct asymptotic r-1 behavior is difficult to achieve. Following van Leeuwen and Baerends, this is linked to the energy εmax of the highest occupied orbital arising from the Kohn-Sham equations. This deficiency is linked also with the poor prediction of mol. polarizabilities. The Becke-Roussel (BR) exchange functional is examd., which is derived by assuming a hydrogen-like exchange hole at all spatial points, and it has the attraction of being dependent on both the kinetic energy d. and the Laplacian of the d. and has no adjustable parameters. Becke has presented encouraging results using this functional in a hybrid manner. Fully self-consistent Kohn-sham calcns. are performed using it in combination with Perdew's 1986 correlation functional. The results are very encouraging indeed, so much so that this exchange functional is the best generalized gradient approxn. (GGA) yet discovered. In particular, bond lengths of many mol. show a substantial improvement over results from other GGAs. For example, many CH bonds are now within exptl. accuracy, instead of being typically 0.02 Å too long. Our ab initio understanding of non-dynamic correlation and dynamic correlation is then linked with d. functional theory. It is argued that correlation functionals should pick up the local dynamic correlation, whereas exchange functionals should include non-dynamic correlation effects. For these reasons it is considered that exchange functionals are best modeled on a system for which there is effectively no non-dynamic correlation, for which the optimum example is the Ne atom. Thus, again following Becke and Roussel, the spherically averaged Hartree-Fock exchange hole for Ne is examd., compared with the BR model functional hole. An excellent overlap is found, and thus the above good results are explained. As a final contribution, the dissocn. of the H2 mol. is re-examd., looking at it in terms of the exchange hole. For a ref. electron near one proton A, the RHF model has half an exchange electron near it, and half the exchange electron near the other proton B, whereas the BR functional has one electron near the other proton b, whereas the Br functional has one electron near A, which is the correct picture. For this reason the (restricted) BR functional gives a greatly improved dissocn. curve for H2 when compared with the Hartree-Fock curve. In summary, the Becke-Roussel functional is found to be a most attractive exchange functional.**68**Pople, J. A.; Gill, P. M.; Johnson, B. G. Kohn-Sham density-functional theory within a finite basis set.*Chem. Phys. Lett.*1992,*199*, 557– 560, DOI: 10.1016/0009-2614(92)85009-YGoogle Scholar68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXhtV2qsrg%253D&md5=8fe4c7a7e9c3f9f7282d3234d343b03cKohn-Sham density-functional theory within a finite basis setPople, John A.; Gill, Peter M. W.; Johnson, Benny G.Chemical Physics Letters (1992), 199 (6), 557-60CODEN: CHPLBC; ISSN:0009-2614.The Kohn-Sham self-consistent equations, using a finite orbital basis expansion, are formulated for exchange-correlation functionals which depend on local densities and their gradients. It is shown that these can be solved iteratively without evaluation of d. Hessians. A general expression is given for the energy gradient (with respect to nuclear motion) after self-consistency has been achieved.**69**Schmidt, G. D.; Liebert, J.; Harris, H. C.; Dahn, C. C.; Leggett, S. K. Discovery of a Highly Magnetic White Dwarf with Strong Carbon Features.*Astrophys. J.*1999,*512*, 916– 919, DOI: 10.1086/306819Google Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXislaqtrk%253D&md5=1d96e31cfa7b0f8fa1c410908163ac45Discovery of a highly magnetic white dwarf with strong carbon featuresSchmidt, Gary D.; Liebert, James; Harris, Hugh C.; Dahn, Conard C.; Leggett, S. K.Astrophysical Journal (1999), 512 (2, Pt. 1), 916-919CODEN: ASJOAB; ISSN:0004-637X. (University of Chicago Press)Systematic follow-up of high proper-motion stars has identified a new cool magnetic white dwarf that displays spectacular absorption bands in the range 4200-6500 Å. The spectrum bears a striking resemblance to that of LP 790-29, a magnetic DQ star dominated by what are apparently Zeeman-shifted Swan bands of C2. However, key differences in the detailed spectra, polarization, and temp. of the two stars indicate that instead LHS 2229 may represent the 1st case of a magnetic peculiar DQ white dwarf, where absorption in the optical is produced by C2H or another C-H compd. Crude arguments suggest that the field strength on LHS 2229 is in the neighborhood of 108 G. B VI photometry proves to be effective in identifying such peculiar stars, since they lie well outside the main white dwarf sequence in a color-color diagram.**70**Reid, I. N.; Liebert, J.; Schmidt, G. D. Discovery of a Magnetic DZ White Dwarf with Zeeman-Split Lines of Heavy Elements.*Astrophys. J.*2001,*550*, L61– L63, DOI: 10.1086/319481Google Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXjsFSjsLY%253D&md5=0c7d19310fec0a907b2ad093e7596634Discovery of a magnetic DZ white dwarf with Zeeman-split lines of heavy elementsReid, I. Neill; Liebert, James; Schmidt, Gary D.Astrophysical Journal (2001), 550 (1, Pt. 2), L61-L63CODEN: ASJOAB; ISSN:0004-637X. (University of Chicago Press)A spectroscopic survey of unstudied Luyten half-second proper-motion stars has resulted in the discovery of an unusual new magnetic white dwarf. LHS 2534 proves to be the first magnetic DZ, showing Zeeman-split Na I and Mg I components, as well as Ca I and Ca II lines for which Zeeman components are blended. The Na I splittings result in a mean surface field strength est. of 1.92 MG. Apart from the magnetic field, LHS 2534 is one of the most heavily blanketed and coolest DZ white dwarfs at Teff ∼ 6000 K.**71**Kawka, A.; Vennes, S.; Ferrario, L.; Paunzen, E. Evidence of enhanced magnetism in cool, polluted white dwarfs.*Mon. Not. R. Astron. Soc.*2019,*482*, 5201– 5210, DOI: 10.1093/mnras/sty3048Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVeltL8%253D&md5=cfe4c129efc8afb3098cc16cb10fb1b9Evidence of enhanced magnetism in cool, polluted white dwarfsKawka, Adela; Vennes, Stephane; Ferrario, Lilia; Paunzen, ErnstMonthly Notices of the Royal Astronomical Society (2019), 482 (4), 5201-5210CODEN: MNRAA4; ISSN:1365-2966. (Oxford University Press)We report the discovery of a new, polluted, magnetic white dwarf in the Luyten survey of high-proper motion stars. High-dispersion spectra of NLTT 7547 reveal a complex heavy element line spectrum in a cool (≈5200 K) hydrogen-dominated atm. showing the effect of a surface averaged field of 163 kG, consistent with a 240 kG centered dipole, although the actual field structure remains uncertain. The abundance pattern shows the effect of accreted material with a distinct magnesium-rich flavor. Combined with earlier identifications, this discovery supports a correlation between the incidence of magnetism in cool white dwarfs and their contamination by heavy elements.**72**Jones, M. D.; Ortiz, G.; Ceperley, D. M. Hartree-Fock studies of atoms in strong magnetic fields.*Phys. Rev. A: At., Mol., Opt. Phys.*1996,*54*, 219– 231, DOI: 10.1103/PhysRevA.54.219Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xkt1OktLc%253D&md5=f071495632dea366d6658f94fbf7c2baHartree-Fock studies of atoms in strong magnetic fieldsJones, Matthew D.; Ortiz, Gerardo; Ceperley, David M.Physical Review A: Atomic, Molecular, and Optical Physics (1996), 54 (1), 219-231CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We present comprehensive calcns. of the electronic structure of selected first-row atoms in uniform magnetic fields of strength ≤1010 G, within a flexible implementation of the Hartree-Fock formalism. Ground-state and low-lying excited-state properties are presented for first-row atoms He, Li, C, and ion H-. We predict and describe a series of ground-state quantum transitions as a function of magnetic field strength. Due to its astrophys. importance, highly excited states of neutral He are also computed. Comparisons are made with previous works, where available.**73**Ivanov, M. V.; Schmelcher, P. Ground state of the carbon atom in strong magnetic fields.*Phys. Rev. A: At., Mol., Opt. Phys.*1999,*60*, 3558– 3568, DOI: 10.1103/PhysRevA.60.3558Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXmvFKlsLs%253D&md5=71b04a3f6753da21a506c3b6d9b0ed3bGround state of the carbon atom in strong magnetic fieldsIvanov, M. V.; Schmelcher, P.Physical Review A: Atomic, Molecular, and Optical Physics (1999), 60 (5), 3558-3568CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The ground and a few excited states of the carbon atom in external uniform magnetic fields are calcd. by means of our two-dimensional mesh Hartree-Fock method for field strengths ranging from zero up to 2.35 × 109 T. With increasing field strength the ground state undergoes six transitions involving seven different electronic configurations which belong to three groups with different spin projections Sz = -1, -2, -3. For weak fields the ground-state configuration arises from the field-free 1s22s22p02p-1, Sz = -1 configuration. With increasing field strength the ground state involves the four Sz = -2 configurations 1s22s2p02p-12p+1, 1s22s2p02p-13d-2, 1s22p02p-13d-24f-3, and 1s22p-13d-24f-35g-4, followed by the two fully spin-polarized Sz = -3 configurations 1s2p02p-13d-24f-35g-4 and 1s2p-13d-24f-35g-46h-5. The last configuration forms the ground state of the carbon atom in the high-field regime γ > 18.664. The above series of ground-state configurations is extd. from the results of numerical calcns. for more than 20 electronic configurations selected due to some general energetic arguments.**74**Ivanov, M. V.; Schmelcher, P. The boron atom and boron positive ion in strong magnetic fields.*J. Phys. B: At., Mol. Opt. Phys.*2001,*34*, 2031– 2044, DOI: 10.1088/0953-4075/34/10/316Google Scholar74https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXktlSltrs%253D&md5=db05a6fe01708387c70da334cc04331aThe boron atom and boron positive ion in strong magnetic fieldsIvanov, M. V.; Schmelcher, P.Journal of Physics B: Atomic, Molecular and Optical Physics (2001), 34 (10), 2031-2044CODEN: JPAPEH; ISSN:0953-4075. (Institute of Physics Publishing)The ground and a few excited states of the B atom in external uniform magnetic fields are calcd. by a 2-dimensional mesh Hartree-Fock method for field strengths ranging from zero up to 2.35 × 109 T. With increasing field strength the ground state of the B atom undergoes five crossovers involving six different electronic configurations which belong to three groups with different spin projections Sz = -1/2, -3/2, -5/2. For weak fields the ground state configuration arises from the field-free 1s22s22p-1, Sz = -1/2 configuration. With increasing field strength the ground state involves the four Sz = -3/2 configurations 1s22s2p02p-1, 1s22s2p-13d-2, 1s22p02p-13d-2 and 1s22p-13d-24f-3, followed by the fully spin-polarized Sz = -5/2 configuration 1s2p-13d-24f-35g-4. The latter configuration forms the ground state of the B atom in the high-field regime γ > 8.0251. Analogous calcns. for the B+ give a sequence of the four following ground state configurations: 1s22s2 (Sz = 0), 1s22s2p-1 (Sz = -1), 1s22p-13d-2 (Sz = -1) and 1s2p-13d-24f-3 (Sz = -2). The above series of ground state configurations are extd. from the results of numerical calcns. for a no. of electronic configurations selected according to general energetical arguments.**75**Al-Hujaj, O.-A.; Schmelcher, P. Lithium in strong magnetic fields.*Phys. Rev. A: At., Mol., Opt. Phys.*2004,*70*, 033411, DOI: 10.1103/PhysRevA.70.033411Google Scholar75https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXovFGisL4%253D&md5=f26437bde2df5dd07fcbeff0d479505fLithium in strong magnetic fieldsAl-Hujaj, Omar-Alexander; Schmelcher, PeterPhysical Review A: Atomic, Molecular, and Optical Physics (2004), 70 (3), 033411/1-033411/12CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The electronic structure of the lithium atom in a strong magnetic field 0 ≤ γ ≤ 10 is investigated. Our computational approach is a full CI method based on a set of anisotropic Gaussian orbitals that is nonlinearly optimized for each field strength. Accurate results for the total energies and one-electron ionization energies for the ground and several excited states for each of the symmetries 20+, 2(-1)+, 4(-1)+, 4(-1)-, 2(-2)+, 4(-2)+, and 4(-3)+ are presented. The behavior of these energies as a function of the field strength is discussed and classified. Transition wavelengths for linear and circular polarized transitions are presented as well.**76**Berdyugina, S. V.; Berdyugin, A. V.; Piirola, V. Molecular Magnetic Dichroism in Spectra of White Dwarfs.*Phys. Rev. Lett.*2007,*99*, 091101, DOI: 10.1103/PhysRevLett.99.091101Google Scholar76https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXpvVGrsbg%253D&md5=05dbd5cbb45d80548a5b35ff8cf16bf7Molecular Magnetic Dichroism in Spectra of White DwarfsBerdyugina, S. V.; Berdyugin, A. V.; Piirola, V.Physical Review Letters (2007), 99 (9), 091101/1-091101/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)The authors present novel calcns. of the magnetic dichroism appearing in mol. bands in the presence of a strong magnetic field, which perturbs the internal structure of the mol. and results in net polarization due to the Paschen-Back effect. Based on that, the authors analyze new spectropolarimetric observations of the cool magnetic He-rich white dwarf G99-37, which shows strongly polarized mol. bands in its spectrum. In addn. to previously known mol. bands of the C2 Swan and CH A-X systems, the authors find a firm evidence for the violet CH B-X bands at 390 nm and C2 Deslandres-d'Azambuja bands at 360 nm. Combining the polarimetric observations with model calcns., the authors deduce a dipole magnetic field of 7.5 ± 0.5 MG with the pos. pole pointing towards the Earth. The developed technique is an excellent tool for studying magnetic fields on cool magnetic stars.**77**Vornanen, T.; Berdyugina, S. V.; Berdyugin, A. V.; Piirola, V. GJ 841B - The Second DQ White Dwarf with Polarized CH Molecular Bands.*Astrophys. J., Lett.*2010,*720*, L52, DOI: 10.1088/2041-8205/720/1/L52Google ScholarThere is no corresponding record for this reference.**78**Detmer, T.; Schmelcher, P.; Diakonos, F. K.; Cederbaum, L. S. Hydrogen molecule in magnetic fields: The ground states of the Σ manifold of the parallel configuration.*Phys. Rev. A: At., Mol., Opt. Phys.*1997,*56*, 1825– 1838, DOI: 10.1103/PhysRevA.56.1825Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlvVems7o%253D&md5=d12ff91c3ddbefc85db16fb64a48861dHydrogen molecule in magnetic fields: the ground states of the Σ manifold of the parallel configurationDetmer, T.; Schmelcher, P.; Diakonos, F. K.; Cederbaum, L. S.Physical Review A: Atomic, Molecular, and Optical Physics (1997), 56 (3), 1825-1838CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The electronic structure of the hydrogen mol. is investigated for the parallel configuration. The ground states of the Σ manifold are studied for ungerade and gerade parity as well as singlet and triplet states covering a broad regime of field strengths from B = 0 up to B = 100 a.u. A variety of interesting phenomena can be obsd. For the 1Σg state we found a monotonous decrease of the equil. distance and a simultaneous increase of the dissocn. energy with growing magnetic-field strength. The 3Σg state is shown to develop an addnl. min. which has no counterpart in field-free space. The 1Σu state shows a monotonous increase in the dissocn. energy with a first increasing and then decreasing internuclear distance of the min. For this state the dissocn. channel is H2 → H- + H+ for magnetic field strengths B .ltorsim. 20 a.u. due to the existence of strongly bound H- states in strong magnetic fields. The repulsive 3Σu state possesses a very shallow van der Waals min. for magnetic-field strengths smaller than 1.0 a.u. within the numerical accuracy of our calcns. The 1Σg and 3Σu states cross as a function of B and the 3Σu state, which is an unbound state, becomes the ground state of the hydrogen mol. in magnetic fields B .ltorsim. 0.2 a.u. This is of particular interest for the existence of mol. hydrogen in the vicinity of white dwarfs. In superstrong fields the ground state is again a strongly bound state, the 3Πu state.**79**Detmer, T.; Schmelcher, P.; Cederbaum, L. S. Hydrogen molecule in a magnetic field: The lowest states of the Π manifold and the global ground state of the parallel configuration.*Phys. Rev. A: At., Mol., Opt. Phys.*1998,*57*, 1767– 1777, DOI: 10.1103/PhysRevA.57.1767Google Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXhtFOqtrc%253D&md5=b6c8f609ab04b2b1c734796488639525Hydrogen molecule in a magnetic field: The lowest states of the Π manifold and the global ground state of the parallel configurationDetmer, T.; Schmelcher, P.; Cederbaum, L. S.Physical Review A: Atomic, Molecular, and Optical Physics (1998), 57 (3), 1767-1777CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The electronic structure of the hydrogen mol. in a magnetic field is investigated for parallel internuclear and magnetic field axes. The lowest states of the Π manifold are studied for spin singlet and triplet (Ms=-1) as well as gerade and ungerade parity for a broad range of field strengths 0≤B≤100 a.u. For both states with gerade parity we observe a monotonic decrease in the dissocn. energy with increasing field strength up to B=0.1 a.u. and metastable states with respect to the dissocn. into two H atoms occur for a certain range of field strengths. For both states with ungerade parity we observe a strong increase in the dissocn. energy with increasing field strength above some crit. field strength Bc. As a major result we det. the transition field strengths for the crossings among the lowest 1Σg, 3Σu, and 3Πu states. The global ground state for B.ltorsim.0.18 a.u. is the strongly bound 1Σg state. The crossings of the 1Σg with the 3Σu and 3Πu state occur at B≈0.18 and B≈0.39 a.u., resp. The transition between the 3Σu and the 3Πu state occurs at B≈12.3 a.u. Therefore, the global ground state of the hydrogen mol. for the parallel configuration is the unbound 3Σu state for 0.18.ltorsim.B.ltorsim.12.3 a.u. The ground state for B⪆12.3 a.u. is the strongly bound 3Πu state. This result is of great relevance to the chem. in the atmospheres of magnetic white dwarfs and neutron stars.**80**Schmelcher, P. Exploring the topology of potential energy surfaces of the H_{2}^{+}ion in the presence of a strong magnetic field.*Int. J. Quantum Chem.*1997,*64*, 553– 560, DOI: 10.1002/(SICI)1097-461X(1997)64:5<553::AID-QUA6>3.0.CO;2-VGoogle Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlsVyjs7k%253D&md5=8772bf0708335e960c94f6f2dc48b6a3Exploring the topology of potential energy surfaces of the H2+ ion in the presence of a strong magnetic fieldSchmelcher, P.International Journal of Quantum Chemistry (1997), 64 (5), 553-560CODEN: IJQCB2; ISSN:0020-7608. (Wiley)We discuss the symmetries, the behavior of the diabatic energy curves, as well as the static aspects of vibronic interaction for diat. mols. in the presence of a strong magnetic field. Our central subject of investigation is the topol. of the adiabatic electronic potential energy surfaces of diat. mols. which are discussed using some selected examples of the surfaces of the H21+ ion. Global equil. configurations corresponding to stable mol. states are found both for the parallel as well as for the perpendicular configurations. For a higher degree of excitation, we observe that the global min. can belong to the lowest possible symmetry of the ion in the presence of a magnetic field. As an example, we discuss the topol. of the 3u potential energy surface.**81**Augustovičová, L. D.; Špirko, V. Zeeman molecular probe for tests of fundamental physical constants.*Mon. Not. R. Astron. Soc.*2020,*494*, 1675– 1680, DOI: 10.1093/mnras/staa792Google Scholar81https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXis1yktr%252FN&md5=b6c781e4cea284d86ddd15e4328931f4Zeeman molecular probe for tests of fundamental physical constantsAugustovicova, Lucie D.; Spirko, VladimirMonthly Notices of the Royal Astronomical Society (2020), 494 (2), 1675-1680CODEN: MNRAA4; ISSN:1365-2966. (Oxford University Press)The impact of the Zeeman effect on the λ-doublet spectra of diat. radicals is analyzed from the point of view of a possible cosmol. variation of the proton-to-electron mass ratio, ν. The actual model calcns. performed for the 2π3/2 and 2π1/2 states of 16OH reveal that the λ-doublet energy levels of diat. radicals can be tuned to degeneracy by means of the Zeeman effect using realistic magnetic fields. Tuning this degeneracy allows for a dramatic enhancement of the relative mass sensitivity coeffs. of the corresponding transitions and for a substantial redn. of their Doppler broadening. Moreover, unlike their field-free counterparts assocd. with the degeneracies arising due to the A ~ 4B situations (A and B being the spin-orbit and rotation const., resp.), the elec. dipole allowed e↔f Zeeman-tuned transitions exhibit favorable intensities, thus evidencing their promising potential.**82**Labzowsky, L. N.; Lozovik, Y. E. Conjugated molecules in strong magnetic fields.*Int. J. Quantum Chem.*1973,*7*, 985– 989, DOI: 10.1002/qua.560070513Google ScholarThere is no corresponding record for this reference.**83**Luzanov, A. V. Magnetism and the biradicaloid character of π-aromatic and antiaromatic systems in a strong magnetic field.*J. Struct. Chem.*2013,*54*, 277– 282, DOI: 10.1134/S0022476613020017Google Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXotlertbg%253D&md5=0f0ba2223a227256210e35affe15976dMagnetism and the biradicaloid character of π-aromatic and antiaromatic systems in a strong magnetic fieldLuzanov, A. V.Journal of Structural Chemistry (2013), 54 (2), 277-282CODEN: JSTCAM; ISSN:0022-4766. (Springer)The previously developed scheme of the full CI for magnetic perturbations of π systems is transformed into a scheme for calcns. in the finite field. It helps create magnetic portraits of mols., reflecting the essentially nonlinear behavior of conjugated systems in a strong field. In particular, possible latent paramagnetism of arom. systems and correspondingly latent diamagnetism of antiarom. ones is easily detected. The degree of the π electron shell openness as well as the singlet-triplet splitting in the field are evaluated. From the data obtained thus in the strong magnetic field an arom. mol. becomes as a rule biradicaloid and nonarom. Accordingly, an antiarom. system dramatically reduces its initial biradicaloid character and thus loses its antiaromaticity.**84**Bates, J. E.; Furche, F. Harnessing the meta-generalized gradient approximation for time-dependent density functional theory.*J. Chem. Phys.*2012,*137*, 164105, DOI: 10.1063/1.4759080Google Scholar84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhsFGksLrF&md5=d2e6a673d974d619dfe23292efc905b1Harnessing the meta-generalized gradient approximation for time-dependent density functional theoryBates, Jefferson E.; Furche, FilippJournal of Chemical Physics (2012), 137 (16), 164105/1-164105/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functionals within the meta-generalized gradient approxn. (MGGA) are widely used for ground-state electronic structure calcns. However, the gauge variance of the kinetic energy d. τ confounds applications of MGGAs to time-dependent systems, excited states, magnetic properties, and states with strong spin-orbit coupling. Becke and Tao used the paramagnetic c.d. to construct a gauge invariant generalized kinetic energy d. ̂τ. We show that τW ≤ ̂τ, where τW is the von Weizsaecker kinetic energy d. of a one-electron system. Thus, replacing τ by ̂τ leads to current-dependent MGGAs (cMGGAs) that are not only gauge invariant but also restore the accuracy of MGGAs in iso-orbital regions for time-dependent and current-carrying states. The current dependence of cMGGAs produces a vector exchange-correlation (XC) potential in the time-dependent adiabatic Kohn-Sham (KS) equations. While MGGA response properties of current-free ground states become manifestly gauge-variant to second order, linear response properties are affected by a new XC kernel appearing in the cMGGA magnetic orbital rotation Hessian. This kernel reflects the first-order coupling of KS orbitals due to changes in the paramagnetic c.d. and has apparently been ignored in previous MGGA response implementations. Inclusion of the current dependence increases total computation times by less than 50%. Benchmark applications to 109 adiabatic excitation energies using the Tao-Perdew-Staroverov-Scuseria (TPSS) MGGA and its hybrid version TPSSh show that cMGGA excitation energies are slightly lower than the MGGA ones on av., but exhibit fewer outliers. Similarly, the optical rotations of 13 small org. mols. show a small but systematic improvement upon inclusion of the magnetic XC kernel. We conclude that cMGGAs should replace MGGAs in all applications involving time-dependent or current-carrying states. (c) 2012 American Institute of Physics.**85**Schlegel, H. B. Optimization of Equilibrium Geometries and Transition Structures.*Advances in Chemical Physics*; John Wiley & Sons, 1987; pp 249– 286, DOI: 10.1002/9780470142936.ch4 .Google ScholarThere is no corresponding record for this reference.**86**Hratchian, H. P.; Schlegel, H. B.*Theory and Applications of Computational Chemistry*; Elsevier, 2005; pp 195– 249.Google ScholarThere is no corresponding record for this reference.**87**Schlegel, H. B. Geometry optimization.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2011,*1*, 790– 809, DOI: 10.1002/wcms.34Google Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtFWrtrbI&md5=62775a44786f69fbe407f459f9652995Geometry optimizationSchlegel, H. BernhardWiley Interdisciplinary Reviews: Computational Molecular Science (2011), 1 (5), 790-809CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. Geometry optimization is an important part of most quantum chem. calcns. This article surveys methods for optimizing equil. geometries, locating transition structures, and following reaction paths. The emphasis is on optimizations using quasi-Newton methods that rely on energy gradients, and the discussion includes Hessian updating, line searches, trust radius, and rational function optimization techniques. Single-ended and double-ended methods are discussed for transition state searches. Single-ended techniques include quasi-Newton, reduced gradient following and eigenvector following methods. Double-ended methods include nudged elastic band, string, and growing string methods. The discussions conclude with methods for validating transition states and following steepest descent reaction paths.**88**Birkholz, A. B.; Schlegel, H. B. Exploration of some refinements to geometry optimization methods.*Theor. Chem. Acc.*2016,*135*, 84, DOI: 10.1007/s00214-016-1847-3Google ScholarThere is no corresponding record for this reference.**89**Baker, J. Techniques for geometry optimization: A comparison of Cartesian and natural internal coordinates.*J. Comput. Chem.*1993,*14*, 1085– 1100, DOI: 10.1002/jcc.540140910Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtlCmsL0%253D&md5=7eefaf59cb2f3d8be6d14bcd924cbea2Techniques for geometry optimization: a comparison of Cartesian and natural internal coordinatesBaker, JonJournal of Computational Chemistry (1993), 14 (9), 1085-100CODEN: JCCHDD; ISSN:0192-8651.A comparison was made between geometry optimization in Cartesian coordinates (using an appropriate initial Hessian) and in natural internal coordinates. Results on 33 different mols., covering a wide range of symmetries and structural types, demonstrated that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internal coordinates to converge to global min.; whereas, Cartesian optimizations converge to the local min. closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internal coordinated are to be preferred over Z-matrix coordinates. General optimization strategies, using internal coordinates and/or Cartesians, are discussed for both unconstrained and constrained optimization.**90**Baker, J.; Chan, F. The location of transition states: A comparison of Cartesian, Z-matrix, and natural internal coordinates.*J. Comput. Chem.*1996,*17*, 888– 904, DOI: 10.1002/(SICI)1096-987X(199605)17:7<888::AID-JCC12>3.0.CO;2-7Google Scholar90https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XitlyqsLo%253D&md5=b9775b2cace6e28c8e91b2c72fe68b1aThe location of transition states: a comparison of Cartesian, Z-matrix, and natural internal coordinatesBaker, Jon; Chan, ForaJournal of Computational Chemistry (1996), 17 (7), 888-904CODEN: JCCHDD; ISSN:0192-8651. (Wiley)A comparison is made between geometry optimization in Cartesian coordinates, in Z-matrix coordinates, and in natural internal coordinates for the location of transition states. In contrast to the situation with min., where all three coordinate systems are of comparable efficiency if a reliable est. of the Hessian matrix is available at the starting geometry, results for 25 different transition states covering a wide range of structural types demonstrate that in practice Z-matrix coordinates are generally superior. For Cartesian coordinates, the commonly used Hessian update schemes are unable to guarantee preservation of the necessary transition state eigenvalue structure, while current algorithms for generating natural internal coordinates may have difficulty handling the distorted geometries assocd. with transition states. The widely used Eigenvector Following (EF) algorithm is shown to be extremely efficient for optimizing transition states.**91**Broyden, C. G. The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations.*IMA J. Appl. Math*1970,*6*, 76– 90, DOI: 10.1093/imamat/6.1.76Google ScholarThere is no corresponding record for this reference.**92**Fletcher, R. A new approach to variable metric algorithms.*Comput. J.*1970,*13*, 317– 322, DOI: 10.1093/comjnl/13.3.317Google ScholarThere is no corresponding record for this reference.**93**Goldfarb, D. A family of variable-metric methods derived by variational means.*Math. Comput.*1970,*24*, 23– 23, DOI: 10.1090/S0025-5718-1970-0258249-6Google ScholarThere is no corresponding record for this reference.**94**Shanno, D. F. Conditioning of quasi-Newton methods for function minimization.*Math. Comput.*1970,*24*, 647– 647, DOI: 10.1090/S0025-5718-1970-0274029-XGoogle ScholarThere is no corresponding record for this reference.**95**Baker, J. Geometry optimization in Cartesian coordinates: Constrained optimization.*J. Comput. Chem.*1992,*13*, 240– 253, DOI: 10.1002/jcc.540130215Google ScholarThere is no corresponding record for this reference.**96**Baker, J.; Bergeron, D. Constrained optimization in Cartesian coordinates.*J. Comput. Chem.*1993,*14*, 1339– 1346, DOI: 10.1002/jcc.540141111Google Scholar96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXhsFCguw%253D%253D&md5=becce2788711e0ef393d3d0c200b555dConstrained optimization in Cartesian coordinatesBaker, Jon; Bergeron, DoreenJournal of Computational Chemistry (1993), 14 (11), 1339-46CODEN: JCCHDD; ISSN:0192-8651.Modifications are made to a previously published algorithm for constrained optimization in Cartesian coordinates (J. Comp. Chem. 13, 240, 1992) to incorporate both fixed and dummy atoms. Std. distance and angle constraints can now be specified with respect to dummy atoms, greatly extending the range of constraints that can be handled. Fixed atoms can be eliminated from the optimization space and so there is no need to calc. their gradients resulting in potentially significant savings of CPU time in ab initio computations. Several examples illustrate the range and versatility of the modified algorithm.**97**Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.*J. Chem. Phys.*1989,*90*, 1007, DOI: 10.1063/1.456153Google Scholar97https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXksVGmtrk%253D&md5=c6cd67a3748dc61692a9cb622d2694a0Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogenDunning, Thom H., Jr.Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.**98**Irons, T. J. P.; Spence, L.; David, G.; Speake, B. T.; Helgaker, T.; Teale, A. M. Analyzing Magnetically Induced Currents in Molecular Systems Using Current-Density-Functional Theory.*J. Phys. Chem. A*2020,*124*, 1321– 1333, DOI: 10.1021/acs.jpca.9b10833Google Scholar98https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhs12ltLc%253D&md5=1b5b59e765d3dc659bc54ad5e300fbc6Analyzing Magnetically Induced Currents in Molecular Systems Using Current-Density-Functional TheoryIrons, Tom J. P.; Spence, Lucy; David, Gregoire; Speake, Benjamin T.; Helgaker, Trygve; Teale, Andrew M.Journal of Physical Chemistry A (2020), 124 (7), 1321-1333CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)A suite of tools for the anal. of magnetically induced currents is introduced. These are applicable to both the weak-field regime, well described by linear response perturbation theory, and to the strong-field regime, which is inaccessible to such methods. A disk-based quadrature scheme is proposed for the anal. of magnetically induced current susceptibilities, providing quadratures that are consistently defined between different mol. systems and applicable to both planar 2D and general 3D mol. systems in a black-box manner. The applicability of the approach is demonstrated for a range of planar ring systems, the ground and excited states of the benzene mol., and the ring, bowl, and cage isomers of the C20 mol. in the presence of a weak magnetic field. In the presence of a strong magnetic field, the para- to diamagnetic transition of the BH mol. is studied, demonstrating that magnetically induced currents present a visual interpretation of this phenomenon, providing insight beyond that accessible using linear response methods.**99**Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations.*J. Chem. Phys.*2002,*116*, 3175– 3183, DOI: 10.1063/1.1445115Google Scholar99https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhtlSiu7k%253D&md5=0130fa656254a693e80d4be6b0f442b8Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculationsWeigend, Florian; Kohn, Andreas; Hattig, ChristofJournal of Chemical Physics (2002), 116 (8), 3175-3183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The convergence of the second-order Moller-Plesset perturbation theory (MP2) correlation energy with the cardinal no. X is investigated for the correlation consistent basis-set series cc-pVXZ and cc-pV(X+d)Z. For the aug-cc-pVXZ and aug-cc-pV(X+d)Z series the convergence of the MP2 correlation contribution to the dipole moment is studied. It is found that, when d-shell electrons cannot be frozen, the cc-pVXZ and aug-cc-pVXZ basis sets converge much slower for third-row elements then they do for first- and second-row elements. Based on the results of these studies criteria are deduced for the accuracy of auxiliary basis sets used in the resoln. of the identity (RI) approxn. for electron repulsion integrals. Optimized auxiliary basis sets for RI-MP2 calcns. fulfilling these criteria are reported for the sets cc-pVXZ, cc-pV(X+d)Z, aug-cc-pVXZ, and aug-cc-pV(X+d)Z with X=D, T, and Q. For all basis sets the RI error in the MP2 correlation energy is more than two orders of magnitude smaller than the usual basis-set error. For the auxiliary aug-cc-pVXZ and aug-cc-pV(X+d)Z sets the RI error in the MP2 correlation contribution to the dipole moment is one order of magnitude smaller than the usual basis set error. Therefore extrapolations towards the basis-set limit are possible within the RI approxn. for both energies and properties. The redn. in CPU time obtained with the RI approxn. increases rapidly with basis set size. For the cc-pVQZ basis an acceleration by a factor of up to 170 is obsd.**100**Jost, R. Magnetic field control of molecular dissociation energies.*Int. J. Quantum Chem.*1997,*64*, 571– 580, DOI: 10.1002/(SICI)1097-461X(1997)64:5<571::AID-QUA8>3.0.CO;2-TGoogle Scholar100https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlsVyis74%253D&md5=8459496e23a9a92a0ad9579374fadbd9Magnetic field control of molecular dissociation energiesJost, RemyInternational Journal of Quantum Chemistry (1997), 64 (5), 571-580CODEN: IJQCB2; ISSN:0020-7608. (Wiley)We show that it is possible to control the dissocn. energies of mols. with an external magnetic field. We focus our interest on the lowest dissocn. channel for which the two at. and/or mol. products are formed in their ground state. The crucial requirement is the paramagnetic character of at least one of the two dissocn. products. Then, an external magnetic field lowers the energy of the paramagnetic species in its lowest Zeeman component and, possibly, the corresponding energy of dissocn. of the parent mol. This it true for diat. mols. when at least one of the atoms has an odd no. of electrons. This is also true for oxygen and phosphorus atoms which have a 3P2 ground state. The Zeeman energy shift of paramagnetic species is always of the order of 1 cm-1 per T. The main theor. difficulty is to det. the correlation diagram existing between the bound states of the parent mol. and the states of the products, or equivalently, how the energy evolves as a function of the internuclear distance corresponding to the dissocn. coordinate. Little is known about this evolution, except for diat. mols., because the large internuclear distances are difficult to observe exptl. The main part of the information come from ab initio calcns. For diat. mols., the dissocn. coordinate is also the unique internuclear distance while for polyat. mols., the potential energy surface has 3N - 6 coordinates and multidimensional effects should be considered. In any case, the singlet-triplet-quintet, etc... (or doublet-quartet, etc...) interactions should play an important role in the correlation diagram because crossings are expected between singlet and triplet potential energy curves (from short to long internuclear distances) and these interactions transform the crossings into anticrossings. The specific examples of alkali diat. mols. (Li2, Na2, etc...), of NO2 and of (O2)2 are analyzed in details.**101**Runge, K.; Sabin, J. R. Electronic properties of H_{2}^{+}, H_{2}, and LiH in high magnetic fields.*Int. J. Quantum Chem.*1997,*64*, 561– 570, DOI: 10.1002/(SICI)1097-461X(1997)64:5<561::AID-QUA7>3.0.CO;2-UGoogle Scholar101https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlsVyjsbw%253D&md5=a328cd139be07817ff9c348526ac0941Electronic properties of H2+, H2, and LiH in high magnetic fieldsRunge, Keith; Sabin, John R.International Journal of Quantum Chemistry (1997), 64 (5), 561-570CODEN: IJQCB2; ISSN:0020-7608. (Wiley)Advances in magnet construction technol. have made magnets available with continuous fields of nearly 50 T and with bores of sufficient diam. for expts. In addn. to these magnets, already in use at the National High Magnetic Field Lab. (NHMFL), semicontinuous pulsed sources of 100 T are anticipated in the near future. At its Los Alamos campus, the NHMFL has detonated pulsed magnets of over 1000 T. It thus becomes possible to investigate the behavior of mols. in strong fields with an eye to field-induced changes in such quantities as geometrical and electronic properties, spectroscopic properties, and reactivities. Theory is a useful probe for these quantities and serves to screen among possible candidates for expts. In this contribution, we report preliminary results on calcns. of electronic properties of H21+, H2, and LiH, the simplest of mols. Initial indications are that for increasing applied field strength, mol. bond lengths decrease and binding energies increase, with a concomitant increase in vibrational frequencies. Field-induced changes in these quantities, as well as in ground-state mol. potential energy surfaces are discussed, and suggestions are made for further investigations, both theor. and exptl.**102**Ceulemans, A. J.*Group Theory Applied to Chemistry*; Springer: Dordrecht, The Netherlands, 2013.Google ScholarThere is no corresponding record for this reference.**103**Lange, K. K.; Tellgren, E. I.; Hoffmann, M. R.; Helgaker, T. A Paramagnetic Bonding Mechanism for Diatomics in Strong Magnetic Fields.*Science*2012,*337*, 327– 331, DOI: 10.1126/science.1219703Google Scholar103https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtVOisrrP&md5=9955432d321e69570d849c93efd064eeA Paramagnetic Bonding Mechanism for Diatomics in Strong Magnetic FieldsLange, Kai K.; Tellgren, E. I.; Hoffmann, M. R.; Helgaker, T.Science (Washington, DC, United States) (2012), 337 (6092), 327-331CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)Elementary chem. distinguishes two kinds of strong bonds between atoms in mols.: the covalent bond, where bonding arises from valence electron pairs shared between neighboring atoms, and the ionic bond, where transfer of electrons from one atom to another leads to Coulombic attraction between the resulting ions. We present a third, distinct bonding mechanism: perpendicular paramagnetic bonding, generated by the stabilization of antibonding orbitals in their perpendicular orientation relative to an external magnetic field. In strong fields such as those present in the atms. of white dwarfs (on the order of 105 teslas) and other stellar objects, our calcns. suggest that this mechanism underlies the strong bonding of H2 in the 3Σu+(1σg1σu*) triplet state and of He2 in the 1Σg+(1σg21σu*2) singlet state, as well as their preferred perpendicular orientation in the external field.**104**Austad, J.; Borgoo, A.; Tellgren, E. I.; Helgaker, T. Bonding in the helium dimer in strong magnetic fields: the role of spin and angular momentum.*Phys. Chem. Chem. Phys.*2020,*22*, 23502– 23521, DOI: 10.1039/D0CP03259JGoogle Scholar104https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhvFKntLfL&md5=a9ba655c43d1edd73b228c1aae357538Bonding in the helium dimer in strong magnetic fields: the role of spin and angular momentumAustad, Jon; Borgoo, Alex; Tellgren, Erik I.; Helgaker, TrygvePhysical Chemistry Chemical Physics (2020), 22 (41), 23502-23521CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We investigate the helium dimer in strong magnetic fields, focusing on the spectrum of low-lying electronic states and their dissocn. curves, at the full configuration-interaction level of theory. To address the loss of cylindrical symmetry and angular momentum as a good quantum no. for nontrivial angles between the bond axis and magnetic field, we introduce the almost quantized angular momentum (AQAM) and show that it provides useful information about states in arbitrary orientations. In general, strong magnetic fields dramatically rearrange the spectrum, with the orbital Zeeman effect bringing down states of higher angular momentum below the states with pure σ character as the field strength increases. In addn., the spin Zeeman effect pushes triplet states below the lowest singlet; in particular, a field of one at. unit is strong enough to push a quintet state below the triplets. In general, the angle between the bond axis and the magnetic field also continuously modulates the degree of σ, π, and δ character of bonds and the previously identified perpendicular paramagnetic bonding mechanism is found to be common among excited states. Electronic states with preferred skew field orientations are identified and rationalized in terms of permanent and induced electronic currents.**105**Chu, S.-I.; Yoshimine, M.; Liu, B. Ab initio study of the*X*^{2}Π and*A*^{2}Σ^{+}states of OH. I. Potential curves and properties.*J. Chem. Phys.*1974,*61*, 5389– 5395, DOI: 10.1063/1.1681891Google Scholar105https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2MXktFGqt70%253D&md5=a0a1fba770c15b08afda9cdef185eefcAb initio study of the X2Π and A2Σ+ states of the hydroxyl radical. I. Potential curves and propertiesChu, Shih-I; Yoshimine, M.; Liu, B.Journal of Chemical Physics (1974), 61 (12), 5389-95CODEN: JCPSA6; ISSN:0021-9606.The CI wave functions, potential-energy curves, and 1-electron properties are presented. The calcd. equil. internuclear sepn. (Re, in bohrs), dissocn. energy (De, in eV), and dipole moment (μ, in D) in the v = 0 vibrational state, resp., are: OH(X2π), 1.841, 4.43, 1.634; OH(A2Σ+), 1.906, 2.29, 1.875. Calcd. values are also given for OD. The spectroscopic consts. for OH and OD calcd. from the theor. potential curves agree satisfactorily with the available exptl. data. The other mol. properties calcd. include the quadrupole moments and the elec. field gradients at the nuclei.**106**Qin, X.; Zhang, S. D. Low-lying electronic states of the OH radical: Potential energy curves, dipole moment functions, and transition probabilities.*J. Korean Phys. Soc.*2014,*65*, 2017– 2022, DOI: 10.3938/jkps.65.2017Google Scholar106https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhs1eiurk%253D&md5=18765e7c1b96714b563a5d02fe850afeLow-lying electronic states of the OH radical: Potential energy curves, dipole moment functions, and transition probabilitiesQin, X.; Zhang, S. D.Journal of the Korean Physical Society (2014), 65 (12), 2017-2022CODEN: JKPSDV; ISSN:0374-4884. (Korean Physical Society)The six doublet and the two quartet electronic states (2Σ+(2), 2Σ-, 2Π(2), 2Δ, 4Σ-, and 4Π) of the OH radical have been studied using the multi-ref. CI (MRCI) method where the Davidson correction, core-valence interaction and relativistic effect are considered with large basis sets of aug-cc-pv5z, aug-cc-pcv5z, and cc-pv5z-DK, resp. Potential energy curves (PECs) and dipole moment functions are also calcd. for these states for internuclear distances ranging from 0.05 nm to 0.80 nm. All possible vibrational levels and rotational consts. for the bound state X2Π and A2Σ+ of OH are predicted by numerical solving the radial Schrodinger equation through the Level program, and spectroscopic parameters, which are in good agreements with exptl. results, are obtained. Transition dipole moments between the ground state X2Π and other excited states are also computed using MRCI, and the transition probability, lifetime, and Franck-Condon factors for the A2Σ+-X2Π transition are discussed and compared with existing exptl. values.**107**Maeda, K.; Wall, M. L.; Carr, L. D. Hyperfine structure of the hydroxyl free radical (OH) in electric and magnetic fields.*New J. Phys.*2015,*17*, 045014, DOI: 10.1088/1367-2630/17/4/045014Google ScholarThere is no corresponding record for this reference.**108**Bartlett, R. J.; Stanton, J. F. Applications of Post-Hartree–Fock Methods: A Tutorial. In*Reviews in Computational Chemistry*; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH: New York, 1994; Vol. 5, pp 65− 169.Google ScholarThere is no corresponding record for this reference.**109**Gilbert, A. T. B.; Besley, N. A.; Gill, P. M. W. Self-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)†.*J. Phys. Chem. A*2008,*112*, 13164– 13171, DOI: 10.1021/jp801738fGoogle Scholar109https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtValurbL&md5=3baaf7b15c1c6fcd86bc3c071deacfadSelf-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)Gilbert, Andrew T. B.; Besley, Nicholas A.; Gill, Peter M. W.Journal of Physical Chemistry A (2008), 112 (50), 13164-13171CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present a simple algorithm, which we call the max. overlap method (MOM), for finding excited-state solns. to SCF equations. Instead of using the aufbau principle, the algorithm maximizes the overlap between the occupied orbitals on successive SCF iterations. This prevents variational collapse to the ground state and guides the SCF process toward the nearest, rather than the lowest energy, soln. The resulting excited-state solns. can be treated in the same way as the ground-state soln. and, in particular, derivs. of excited-state energies can be computed using ground-state code. We assess the performance of our method by applying it to a variety of excited-state problems including the calcn. of excitation energies, charge-transfer states, and excited-state properties.**110**Besley, N. A.; Gilbert, A. T. B.; Gill, P. M. W. Self-consistent-field calculations of core excited states.*J. Chem. Phys.*2009,*130*, 124308, DOI: 10.1063/1.3092928Google Scholar110https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXjvVemu7c%253D&md5=2b14b2b3b6f825581ee600b9872e278cSelf-consistent-field calculations of core excited statesBesley, Nicholas A.; Gilbert, Andrew T. B.; Gill, Peter M. W.Journal of Chemical Physics (2009), 130 (12), 124308/1-124308/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The accuracy of core excitation energies and core electron binding energies computed within a Δself-consistent-field framework is assessed. The variational collapse of the core excited state is prevented by maintaining a singly occupied core orbital using an overlap criterion called the max. overlap method. When applied to a wide range of small org. mols., the resulting core excitation energies are not systematically underestimated as obsd. in time-dependent d. functional theory and agree well with expt. The accuracy of this approach for core excited states is illustrated by the calcn. of the pre-edge features in x-ray absorption spectra of plastocyanin, which shows that accurate results can be achieved with Δself-consistent-field calcns. when used in conjunction with uncontracted basis functions. (c) 2009 American Institute of Physics.**111**Barca, G. M. J.; Gilbert, A. T. B.; Gill, P. M. W. Simple Models for Difficult Electronic Excitations.*J. Chem. Theory Comput.*2018,*14*, 1501– 1509, DOI: 10.1021/acs.jctc.7b00994Google Scholar111https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXivVaht7c%253D&md5=ca2b971e3071f917108202889deb5addSimple Models for Difficult Electronic ExcitationsBarca, Giuseppe M. J.; Gilbert, Andrew T. B.; Gill, Peter M. W.Journal of Chemical Theory and Computation (2018), 14 (3), 1501-1509CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a single-determinant approach to three challenging topics in the chem. of excited states: double excitations, charge-transfer states, and conical intersections. The results are obtained by using the Initial Maximum Overlap Method (IMOM) which is a modified version of the Maximum Overlap Method (MOM). The new algorithm converges better than the original, esp. for these difficult problems. By considering several case studies, we show that a single-determinant framework provides a simple and accurate alternative for modeling excited states in cases where other low-cost methods, such as CIS and TD-DFT, either perform poorly or fail completely.**112**Burton, H. G. A.; Thom, A. J. W. Holomorphic Hartree-Fock Theory: An Inherently Multireference Approach.*J. Chem. Theory Comput.*2016,*12*, 167– 173, DOI: 10.1021/acs.jctc.5b01005Google Scholar112https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvVOrt77P&md5=94c0f990e9c94212c29b68ebb6ec078fHolomorphic Hartree-Fock Theory: An Inherently Multireference ApproachBurton, Hugh G. A.; Thom, Alex J. W.Journal of Chemical Theory and Computation (2016), 12 (1), 167-173CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We investigate the existence of holomorphic Hartree-Fock solns. using a revised SCF algorithm. We use this algorithm to study the Hartree-Fock solns. for H2 and H42+ and report the emergence of holomorphic solns. at points of symmetry breaking. Finally, we find these holomorphic solns. for H4 and use them as a basis for Non-Orthogonal CI at a range of rectangular geometries and show them to produce energies in good agreement with Full CI.**113**Baerends, E.; Branchadell, V.; Sodupe, M. Atomic reference energies for density functional calculations.*Chem. Phys. Lett.*1997,*265*, 481– 489, DOI: 10.1016/S0009-2614(96)01449-2Google Scholar113https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXnvVKnsw%253D%253D&md5=fe2d05e516372a960660c19551bd95f0Atomic reference energies for density functional calculationsBaerends, E. J.; Branchadell, V.; Sodupe, M.Chemical Physics Letters (1997), 265 (3-5), 481-489CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)At. ground states are usually degenerate. It is demonstrated that the d. functionals for the exchange-correlation energy that are commonly used are not invariant over the set of ground state densities. This leads to uncertainties of the order of 3 to 5 kcal/mol in the at. ground state energy of second and third period main group elements and the first transition series. A much larger spread in energies is obtained for transition elements if symmetry and equivalence restrictions for the Kohn-Sham orbitals are abandoned. It is recommended that at. ground states that are actually used to calc. heats of atomization are made explicit, and tables with one choice of at. ground state energies for the first rows of the periodic system are provided for the local d. approxn. and for a few generalized gradient approxns.**114**Caputo, M. C.; Lazzeretti, P. Geometry distortion of the benzene molecule in a strong magnetic field.*Int. J. Quantum Chem.*2011,*111*, 772– 779, DOI: 10.1002/qua.22812Google Scholar114https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhsF2mt77E&md5=e51b639aa968e3bc6260f96ca42f577eGeometry distortion of the benzene molecule in a strong magnetic fieldCaputo, M. C.; Lazzeretti, P.International Journal of Quantum Chemistry (2011), 111 (4), 772-779CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The electrostatic Lorentz force acting on the H and C nuclei of a benzene mol. in the presence of a strong magnetic field with flux d. B has been estd. via Rayleigh-Schrodinger perturbation theory to second order in B. In stationary conditions, a new equil. configuration is reached, at which the total force has been entirely transferred to the nuclei, and the force on the electrons vanishes. The distortion of the mol. geometry is rationalized in terms of third-rank elec. hypershielding at the nuclei, induced by strong magnetic fields applied along three Cartesian axes. The nuclear hypershielding has been evaluated at near Hartree-Fock level of accuracy by its definition within the Rayleigh-Schrodinger perturbation theory, and by a pointwise procedure for the geometrical derivs. of magnetic susceptibilities. The connection between these two quantities is provided by the Hellmann-Feynman theorem. A field along the C6 symmetry axis causes a sym. contraction of the carbon ring and an elongation of the CH bonds. A field along one of the C2 symmetry axes contg. two CH bond acts to shorten them, to widen the ring, and to bend the four remaining CH bonds towards C2. A field along one of the C symmetry axes through the midpoint of two opposite CC bonds causes a spindle effect, by squeezing the mol. toward the center of mass. Constraints for rotational and translational invariance and hypervirial theorems provide a natural criterion for Hartree-Fock quality of computed nuclear elec. hypershielding. However, the mol. distortion is negligible for applied fields usually available in a lab. ⊗ 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011.**115**Kuchitsu, T.; Okuda, J.; Tachikawa, M. Evaluation of molecular integral of Cartesian Gaussian type basis function with complex-valued center coordinates and exponent via the McMurchie-Davidson recursion formula and its application to electron dynamics.*Int. J. Quantum Chem.*2009,*109*, 540– 548, DOI: 10.1002/qua.21813Google ScholarThere is no corresponding record for this reference.**116**Kawashima, Y.; Ishimura, K.; Shiga, M. Ab initio quantum mechanics/molecular mechanics method with periodic boundaries employing Ewald summation technique to electron-charge interaction: Treatment of the surface-dipole term.*J. Chem. Phys.*2019,*150*, 124103, DOI: 10.1063/1.5048451Google Scholar116https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXmtVClt7s%253D&md5=1b04a92a4d34298bb2e87817f18239bfAb initio quantum mechanics/molecular mechanics method with periodic boundaries employing Ewald summation technique to electron-charge interaction: Treatment of the surface-dipole termKawashima, Y.; Ishimura, K.; Shiga, M.Journal of Chemical Physics (2019), 150 (12), 124103/1-124103/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have developed a combined quantum mechanics/mol. mechanics (QM/MM) method with periodic boundary condition (PBC) treatment of explicit electron-charge interactions in a theor. rigorous manner, for an accurate description of electronic structures for mols. in the condensed phase. The Ewald summation technique is employed for the calcn. of the one-electron Hamiltonian in an ab initio framework. We decomp. the Coulomb interactions into two components: those within the same cell and those between different cells. The former is calcd. in the same way as the conventional QM/MM calcn. for isolated systems; this article focuses on our novel method for calcg. the latter type of Coulomb interactions. The detailed formulation of the Hamiltonian of this new QM/MM-PBC method, as well as the necessary one-electron integrals and their gradients, is given. The novel method is assessed by applying it to the dil. water system and a system with a coumarin mol. in water solvent; it successfully reproduces the electronic energies, frontier orbital energies, and Mulliken population charge of the real-space limit calcd. by QM/MM using large isolated systems. We investigated the contribution from each term of the Hamiltonian and found that the surface-dipole term in the Ewald summation technique is indispensable for QM/MM-PBC calcns. The newly developed QM/MM-PBC method is promising for tackling chem. reactions and excited states of mols. in the condensed phase. (c) 2019 American Institute of Physics.**117**Flocke, N. On the use of shifted Jacobi polynomials in accurate evaluation of roots and weights of Rys polynomials.*J. Chem. Phys.*2009,*131*, 064107, DOI: 10.1063/1.3204437Google Scholar117https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtVSiu7zI&md5=37a6d3d47835cd831b8e17ad2e42bfc7On the use of shifted Jacobi polynomials in accurate evaluation of roots and weights of Rys polynomialsFlocke, N.Journal of Chemical Physics (2009), 131 (6), 064107/1-064107/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Shifted Jacobi polynomials Gn(p,q,x) can be used in connection with the Gaussian quadrature modified moment technique to greatly enhance the accuracy of evaluation of Rys roots and wts. used in Gaussian integral evaluation in quantum chem. A general four-term inhomogeneous recurrence relation is derived for the shifted Jacobi polynomial modified moments over the Rys wt. function e-Tx/x. It is shown that for q = 1/2 this general four-term inhomogeneous recurrence relation reduces to a three-term p-dependent inhomogeneous recurrence relation. Adjusting p to proper values depending on the Rys exponential parameter T, the method is capable of delivering highly accurate results for large no. of roots and wts. in the most difficult to treat intermediate T range. Examples are shown, and detailed formulas together with practical suggestions for their efficient implementation are also provided. (c) 2009 American Institute of Physics.

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## References

ARTICLE SECTIONSThis article references 117 other publications.

**1**Tellgren, E. I.; Soncini, A.; Helgaker, T. Nonperturbative*ab initio*calculations in strong magnetic fields using London orbitals.*J. Chem. Phys.*2008,*129*, 154114, DOI: 10.1063/1.2996525Google Scholar1https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtlSmtL%252FM&md5=f962bd8417bc0beb0479ab5299a125a1Nonperturbative ab initio calculations in strong magnetic fields using London orbitalsTellgren, Erik I.; Soncini, Alessandro; Helgaker, TrygveJournal of Chemical Physics (2008), 129 (15), 154114/1-154114/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A SCF (SCF) London-orbital computational scheme to perform gauge-origin independent nonperturbative calcns. for mols. in strong magnetic fields is presented. The crucial difference in the proposed approach with respect to common-origin finite-field SCF implementations consists in the evaluation of mol. integrals over the field-dependent mol. basis functions, which is tantamount to computing mol. integrals in a hybrid Gaussian and plane-wave basis set. The implementation of a McMurchie-Davidson scheme for the calcn. of the mol. integrals over London orbitals is discussed, and preliminary applications of the newly developed code to the calcn. of fourth-rank hypermagnetizabilities for a set of small mols., benzene, and cyclobutadiene are presented. The nonperturbative approach is particularly useful for studying the highly nonlinear response of paramagnetic closed-shell systems such as boron monohydride, or the π-electron response of cyclobutadiene. (c) 2008 American Institute of Physics.**2**Tellgren, E. I.; Helgaker, T.; Soncini, A. Non-perturbative magnetic phenomena in closed-shell paramagnetic molecules.*Phys. Chem. Chem. Phys.*2009,*11*, 5489, DOI: 10.1039/b822262bGoogle Scholar2https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXns1SrsLo%253D&md5=38de45e1e72ab72ba1850b706cc95c05Non-perturbative magnetic phenomena in closed-shell paramagnetic moleculesTellgren, Erik I.; Helgaker, Trygve; Soncini, AlessandroPhysical Chemistry Chemical Physics (2009), 11 (26), 5489-5498CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)By means of non-perturbative ab initio calcns., it is shown that paramagnetic closed-shell mols. are characterized by a strongly nonlinear magnetic response, whose main feature consists of a paramagnetic-to-diamagnetic transition in a strong magnetic field. The phys. origin of this phenomenon is rationalized on the basis of an anal. model based on MO theory. For the largest mols. considered here, the acepleiadylene dianion and the corannulene dianion, the transition field is of the order of 103 T, about one order of magnitude larger than the magnetic field strength currently achievable in exptl. settings. However, our simple model suggests that the paramagnetic-to-diamagnetic transition is a universal property of paramagnetic closed-shell systems in strong magnetic fields, provided no singlet-triplet level crossing occurs for fields smaller than the crit. transition field. Accordingly, fields weaker than 100 T should suffice to trigger the predicted transition for systems whose size is still well within the (medium-large) mol. domain, such as hypothetical antiarom. rings with less than one hundred carbon atoms.**3**Tellgren, E. I.; Reine, S. S.; Helgaker, T. Analytical GIAO and hybrid-basis integral derivatives: application to geometry optimization of molecules in strong magnetic fields.*Phys. Chem. Chem. Phys.*2012,*14*, 9492, DOI: 10.1039/c2cp40965hGoogle Scholar3https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XosVWmu7k%253D&md5=7005cd1fc6f0f2e54a83e82b5401c1b2Analytical GIAO and hybrid-basis integral derivatives: application to geometry optimization of molecules in strong magnetic fieldsTellgren, Erik I.; Reine, Simen S.; Helgaker, TrygvePhysical Chemistry Chemical Physics (2012), 14 (26), 9492-9499CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Anal. integral evaluation is a central task of modern quantum chem. Here we present a general method for evaluating differentiated integrals over std. Gaussian and mixed Gaussian/plane-wave hybrid orbitals. The main idea is to have a representation of basis sets that is flexible enough to enable differentiated integrals to be reinterpreted as std. integrals over modified basis functions. As an illustration of the method, we report a very simple implementation of Hartree-Fock level geometrical derivs. in finite magnetic fields for gauge-origin independent AOs, within the London program. As a quantum-chem. application, we optimize the structure of helium clusters and some well-known covalently bound mols. (water, ammonia and benzene) subject to strong magnetic fields.**4**Tellgren, E. I.; Teale, A. M.; Furness, J. W.; Lange, K. K.; Ekström, U.; Helgaker, T. Non-perturbative calculation of molecular magnetic properties within current-density functional theory.*J. Chem. Phys.*2014,*140*, 034101, DOI: 10.1063/1.4861427Google Scholar4https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXps1emtw%253D%253D&md5=76d696a52c036086a2ad413ec0ee722bNon-perturbative calculation of molecular magnetic properties within current-density functional theoryTellgren, E. I.; Teale, A. M.; Furness, J. W.; Lange, K. K.; Ekstroem, U.; Helgaker, T.Journal of Chemical Physics (2014), 140 (3), 034101/1-034101/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a novel implementation of Kohn-Sham d.-functional theory using London AOs as basis functions. External magnetic fields are treated nonperturbatively, which enable the study of both magnetic response properties and the effects of strong fields, using either std. d. functionals or current-d. functionals-the implementation is the 1st fully self-consistent implementation of the latter for mols. Pilot applications are presented for the finite-field calcn. of mol. magnetizabilities, hypermagnetizabilities, and NMR shielding consts., focusing on the impact of current-d. functionals on the accuracy of the results. Existing current-d. functionals based on the gauge-invariant vorticity are tested and are sensitive to numerical details of their implementation. Also, when appropriately regularized, the resulting magnetic properties show no improvement over std. d.-functional results. An advantage of the present implementation is the ability to apply d.-functional theory to mols. in very strong magnetic fields, where the perturbative approach breaks down. Comparison with high accuracy full-configuration-interaction results show that the inadequacies of current-d. approxns. are exacerbated with increasing magnetic field strength. Std. d.-functionals remain well behaved but fail to deliver high accuracy. The need for improved current-dependent d.-functionals, and how they may be tested using the presented implementation, is discussed in light of the findings. (c) 2014 American Institute of Physics.**5**Sen, S.; Lange, K. K.; Tellgren, E. I. Excited States of Molecules in Strong Uniform and Nonuniform Magnetic Fields.*J. Chem. Theory Comput.*2019,*15*, 3974– 3990, DOI: 10.1021/acs.jctc.9b00103Google Scholar5https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXpvFKgtr8%253D&md5=4f2b2f1fa578beddb5b86d7db58ee795Excited States of Molecules in Strong Uniform and Nonuniform Magnetic FieldsSen, Sangita; Lange, Kai K.; Tellgren, Erik I.Journal of Chemical Theory and Computation (2019), 15 (7), 3974-3990CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)This paper reports an implementation of Hartree-Fock linear response with complex orbitals for computing electronic spectra of mols. in strong external magnetic fields. The implementation is completely general, allowing for spin-restricted, spin-unrestricted, and general two-component ref. states. The method is applied to small mols. placed in strong uniform and nonuniform magnetic fields of astrochem. importance at the RPA level of theory. For uniform fields, where comparison is possible, the spectra are found to be qual. similar to those recently obtained with equation of motion coupled cluster theory. We also study the behavior of spin-forbidden excitations with progressive loss of spin symmetry induced by nonuniform magnetic fields. Finally, the equivalence of length and velocity gauges for oscillator strengths when using complex orbitals is investigated and found to hold numerically.**6**Sun, S.; Williams-Young, D. B.; Stetina, T. F.; Li, X. Generalized Hartree–Fock with Nonperturbative Treatment of Strong Magnetic Fields: Application to Molecular Spin Phase Transitions.*J. Chem. Theory Comput.*2019,*15*, 348– 356, DOI: 10.1021/acs.jctc.8b01140Google Scholar6https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXitlKmsbvJ&md5=573099c8f2198fc79ab35e36861e8e99Generalized Hartree-Fock with Nonperturbative Treatment of Strong Magnetic Fields: Application to Molecular Spin Phase TransitionsSun, Shichao; Williams-Young, David B.; Stetina, Torin F.; Li, XiaosongJournal of Chemical Theory and Computation (2019), 15 (1), 348-356CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a framework of an ab initio variational approach to effectively explore electronic spin phase transitions in mol. systems inside of a homogeneous magnetic field. In order to capture this phenomenon, the complex generalized Hartree-Fock (C-GHF) method is used in the spinor formalism with London orbitals. Recursive algorithms for computing the one- and two-electron integrals of London orbitals are also provided. A Pauli matrix representation of the C-GHF method is introduced to sep. spin contributions from the scalar part of the Fock matrix. Next, spin phase transitions in two different mol. systems are investigated in the presence of a strong magnetic field. Noncollinear spin configurations are obsd. during the spin phase transitions in H2 and a dichromium complex, [(H3N)4Cr(OH)2Cr(NH3)4]4+, with an increase in magnetic field strength. The competing driving forces of exchange coupling and the spin Zeeman effect have been shown to govern the spin phase transition and its transition rate. Addnl., the energetic contributions of the spin Zeeman, orbital Zeeman, and diamagnetic terms to the potential energy surface are also analyzed.**7**Sun, S.; Williams-Young, D.; Li, X. An ab Initio Linear Response Method for Computing Magnetic Circular Dichroism Spectra with Nonperturbative Treatment of Magnetic Field.*J. Chem. Theory Comput.*2019,*15*, 3162– 3169, DOI: 10.1021/acs.jctc.9b00095Google Scholar7https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXms1Wrsbw%253D&md5=ad8cc260678c2169d056bbb8d164a7beAn ab Initio Linear Response Method for Computing Magnetic Circular Dichroism Spectra with Nonperturbative Treatment of Magnetic FieldSun, Shichao; Williams-Young, David; Li, XiaosongJournal of Chemical Theory and Computation (2019), 15 (5), 3162-3169CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Magnetic CD (MCD) expts. provide a sensitive tool for exploring geometric, magnetic, and electronic properties of chem. complexes and condensed matter systems. They are also challenging to simulate because of the need to simultaneously treat the perturbations of a finite magnetic field as well as an optical field. In this work, we introduce an ab initio approach that treats the external magnetic field nonperturbatively with London orbitals for simulating the MCD spectra of closed-shell systems. Effects of a magnetic field are included variationally in the spin-free nonrelativistic Hamiltonian, followed by a linear response formalism to directly calc. the difference in absorption between the left and right circularly polarized light. In addn. to the presentation of underlying math. formalism and implementation, the method developed in this paper has been applied to simulations of MCD spectra of the sodium anion, 2,2,6,6-tetramethylcyclohexanone, and 3-methyl-2-hexanone. Results are discussed and compared to expts.**8**Sun, S.; Beck, R. A.; Williams-Young, D.; Li, X. Simulating Magnetic Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory in Gauge Including Atomic Orbitals.*J. Chem. Theory Comput.*2019,*15*, 6824– 6831, DOI: 10.1021/acs.jctc.9b00632Google Scholar8https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvFSjurzN&md5=0f9ecf7c77e3c6bf8d1ac46c43fdf95aSimulating Magnetic Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory in Gauge Including Atomic OrbitalsSun, Shichao; Beck, Ryan A.; Williams-Young, David; Li, XiaosongJournal of Chemical Theory and Computation (2019), 15 (12), 6824-6831CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Magnetic CD (MCD) spectra are able to provide insights to the geometric, electronic, and magnetic properties of chem. systems. However, they can be challenging to understand and simulate given the need to simultaneously treat both the finite magnetic and optical fields. Thus, efficient simulations are desired to understand the spectra and resolve the mol. electronic states. Real-time dynamics are used widely in the simulation of electronic spectroscopies such as absorption as well as electronic CD, but simulating MCD with real-time dynamics is tech. and theor. challenging. In this work, we introduce a real-time dynamics based ab initio method with a non-perturbative treatment of a static magnetic field with London orbitals for simulating the MCD spectra of closed-shell systems. Effects of a magnetic field are included variationally in the spin-free non-relativistic Hamiltonian. Real-time time dependent d. functional theory dynamics are then performed, from which we compute the response function in the presence of the external magnetic field, giving the MCD spectrum. The method developed in this paper is applied to simulate the MCD spectra for pyrimidine, pyrazine, and 1,4-naphthoquinone. Results are discussed and compared to expt.**9**Stopkowicz, S.; Gauss, J.; Lange, K. K.; Tellgren, E. I.; Helgaker, T. Coupled-cluster theory for atoms and molecules in strong magnetic fields.*J. Chem. Phys.*2015,*143*, 074110, DOI: 10.1063/1.4928056Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhtlOgsrrN&md5=803b9a5e5228c809f7ec5967770cb29cCoupled-cluster theory for atoms and molecules in strong magnetic fieldsStopkowicz, Stella; Gauss, Jurgen; Lange, Kai K.; Tellgren, Erik I.; Helgaker, TrygveJournal of Chemical Physics (2015), 143 (7), 074110/1-074110/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An implementation of coupled-cluster (CC) theory to treat atoms and mols. in finite magnetic fields is presented. The main challenges for the implementation stem from the magnetic-field dependence in the Hamiltonian, or, more precisely, the appearance of the angular momentum operator, due to which the wave function becomes complex and which introduces a gauge-origin dependence. For this reason, an implementation of a complex CC code is required together with the use of gauge-including AOs to ensure gauge-origin independence. Results of coupled-cluster singles-doubles-perturbative-triples (CCSD(T)) calcns. are presented for atoms and mols. with a focus on the dependence of correlation and binding energies on the magnetic field. (c) 2015 American Institute of Physics.**10**Hampe, F.; Stopkowicz, S. Equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fields.*J. Chem. Phys.*2017,*146*, 154105, DOI: 10.1063/1.4979624Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXmtlKhsLk%253D&md5=15e2462d7406ee3e6fa60a14f3b2c9e2Equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fieldsHampe, Florian; Stopkowicz, StellaJournal of Chemical Physics (2017), 146 (15), 154105/1-154105/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A program for the direct calcn. of excitation energies of atoms and mols. in strong magnetic fields is presented. The implementation includes the equation-of-motion coupled-cluster singles-doubles (EOM-CCSD) method for electronically excited states as well as its spin-flip variant. Differences to regular EOM-CCSD implementations are due to the appearance of the canonical angular-momentum operator in the Hamiltonian causing the wave function to become complex. The gauge-origin problem is treated by the use of gauge-including AOs. Therefore, a modified Davidson method for diagonalizing complex non-Hermitian matrixes is used. Excitation energies for selected atoms and mols. that are of importance in the astrochem. context are presented and their dependence on the magnetic field is discussed. (c) 2017 American Institute of Physics.**11**Hampe, F.; Gross, N.; Stopkowicz, S. Full triples contribution in coupled-cluster and equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fields.*Phys. Chem. Chem. Phys.*2020,*22*, 23522– 23529, DOI: 10.1039/D0CP04169FGoogle Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXitFGqu7vO&md5=792d3a16d609d1d13c9886adc6292909Full triples contribution in coupled-cluster and equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fieldsHampe, Florian; Gross, Niklas; Stopkowicz, StellaPhysical Chemistry Chemical Physics (2020), 22 (41), 23522-23529CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Coupled-cluster as well as equation-of-motion coupled-cluster methods play an important role whenever high accuracy is warranted. Concerning excitation energies, consideration of triple excitations is typically required to reach an accuracy better than 0.1-0.3 eV. In the context of strong magnetic fields such accuracy is needed for the prediction of spectra of strongly magnetized White Dwarfs. In addn. it turns out that in order to correctly model the behavior of energies with respect to the magnetic field strength, triple excitations are required. Due to avoided crossings which are extremely often encountered in the context of strong magnetic fields, double-excitation character can be transferred between electronic states of the same symmetry. We report an implementation of the full finite-field coupled-cluster with single, double, and triple substitutions (CCSDT) and the equation-of-motion-CCSDT models and apply them to the prediction of field-dependent transition wavelengths for sodium as well as to the four lowest singlet states of the CH+ mol. in a strong magnetic field.**12**Furness, J. W.; Verbeke, J.; Tellgren, E. I.; Stopkowicz, S.; Ekström, U.; Helgaker, T.; Teale, A. M. Current Density Functional Theory Using Meta-Generalized Gradient Exchange-Correlation Functionals.*J. Chem. Theory Comput.*2015,*11*, 4169– 4181, DOI: 10.1021/acs.jctc.5b00535Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXht1OqsLbJ&md5=bd9f63dc323d39de7ddeccd9a85fe7bbCurrent Density Functional Theory Using Meta-Generalized Gradient Exchange-Correlation FunctionalsFurness, James W.; Verbeke, Joachim; Tellgren, Erik I.; Stopkowicz, Stella; Ekstrom, Ulf; Helgaker, Trygve; Teale, Andrew M.Journal of Chemical Theory and Computation (2015), 11 (9), 4169-4181CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present the self-consistent implementation of current-dependent (hybrid) meta-generalized gradient approxn. (mGGA) d. functionals using London AOs. A previously proposed generalized kinetic energy d. is utilized to implement mGGAs in the framework of Kohn-Sham c.d. functional theory (KS-CDFT). A unique feature of the nonperturbative implementation of these functionals is the ability to seamlessly explore a wide range of magnetic fields up to 1 au (∼235 kT) in strength. CDFT functionals based on the TPSS and B98 forms are investigated, and their performance is assessed by comparison with accurate coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) data. In the weak field regime, magnetic properties such as magnetizabilities and NMR shielding consts. show modest but systematic improvements over generalized gradient approxns. (GGA). However, in the strong field regime, the mGGA-based forms lead to a significantly improved description of the recently proposed perpendicular paramagnetic bonding mechanism, comparing well with CCSD(T) data. In contrast to functionals based on the vorticity, these forms are found to be numerically stable, and their accuracy at high field suggests that the extension of mGGAs to CDFT via the generalized kinetic energy d. should provide a useful starting point for further development of CDFT approxns.**13**Reimann, S.; Borgoo, A.; Austad, J.; Tellgren, E. I.; Teale, A. M.; Helgaker, T.; Stopkowicz, S. Kohn–Sham energy decomposition for molecules in a magnetic field.*Mol. Phys.*2019,*117*, 97– 109, DOI: 10.1080/00268976.2018.1495849Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhtlGhsrjL&md5=daf402b362574e5f792c3ac5acd3e0e6Kohn-Sham energy decomposition for molecules in a magnetic fieldReimann, Sarah; Borgoo, Alex; Austad, Jon; Tellgren, Erik I.; Teale, Andrew M.; Helgaker, Trygve; Stopkowicz, StellaMolecular Physics (2019), 117 (1), 97-109CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)We study the total mol. electronic energy and its Kohn-Sham components within the framework of magnetic-field d.-functional theory (BDFT), an alternative to current-dependent d.-functional theory (CDFT) for mols. in the presence of magnetic fields. For a selection of closed-shell dia- and paramagnetic mols., we investigate the dependence of the total electronic energy and its Kohn-Sham components on the magnetic field. Results obtained from commonly used d.-functional approxns. are compared with those obtained from Lieb optimisations based on magnetic-field dependent relaxed coupled-cluster singles-and-doubles (CCSD) and second-order Moller-Plesset (MP2) densities. We show that popular approx. exchange-correlation functionals at the generalised-gradient-approxn. (GGA), meta-GGA, and hybrid levels of theory provide a good qual. description of the electronic energy and its Kohn-Sham components in a magnetic field-in particular, for the diamagnetic mols. The performance of Hartree-Fock theory, MP2 theory, CCSD theory and BDFT with different exchange-correlation functionals is compared with coupled-cluster singles-doubles-perturbative-triples (CCSD(T)) theory for the perpendicular component of the magnetisability. Generalisations of the TPSS meta-GGA functional to systems in a magnetic field work well-the cTPSS functional, in particular, with a current-cor. kinetic-energy d., performs excellently, providing an accurate and balanced treatment of dia- and paramagnetic systems and outperforming MP2 theory.**14**Lehtola, S.; Dimitrova, M.; Sundholm, D. Fully numerical electronic structure calculations on diatomic molecules in weak to strong magnetic fields.*Mol. Phys.*2020,*118*, e1597989, DOI: 10.1080/00268976.2019.1597989Google ScholarThere is no corresponding record for this reference.**15**Irons, T. J. P.; Zemen, J.; Teale, A. M. Efficient Calculation of Molecular Integrals over London Atomic Orbitals.*J. Chem. Theory Comput.*2017,*13*, 3636– 3649, DOI: 10.1021/acs.jctc.7b00540Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtFChsb7L&md5=dbec737cdb32d62c96bf2c20f35ef389Efficient Calculation of Molecular Integrals over London Atomic OrbitalsIrons, Tom J. P.; Zemen, Jan; Teale, Andrew M.Journal of Chemical Theory and Computation (2017), 13 (8), 3636-3649CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The use of London AOs (LAOs) in a nonperturbative manner enables the detn. of gauge-origin invariant energies and properties for mol. species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calcns. for mol. systems is the evaluation of mol. integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiencies of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP), and Rys quadrature schemes are compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and wts., enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalized algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta; thus, a simple mixed scheme is put forward that selects the most efficient approach to calc. the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm.**16**David, G.; Irons, T. J. P.; Fouda, A. E. A.; Furness, J. W.; Teale, A. M. SCF Methods for Excited States in Strong Magnetic Fields. Manuscript in preparation, 2021.Google ScholarThere is no corresponding record for this reference.**17**Wibowo, M.; Irons, T. J. P.; Teale, A. M. Modelling ultrafast electron dynamics in strong magnetic fields using real-time time-dependent electronic structure methods.*J. Chem. Theory Comput.*2021, DOI: 10.1021/acs.jctc.0c01269 .Google ScholarThere is no corresponding record for this reference.**18**Angel, J. R. P. Magnetism in white dwarfs.*Astrophys. J.*1977,*216*, 1, DOI: 10.1086/155436Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2sXltV2ht7w%253D&md5=738df5f09f6429a0a61fe1af083fbf67Magnetism in white dwarfsAngel, J. R. P.Astrophysical Journal (1977), 216 (1, Pt. 1), 1-17CODEN: ASJOAB; ISSN:0004-637X.A few percent of all white dwarfs are strongly magnetic. Ten examples are now known, of which half have spectra which show Zeeman splitting in lines of H, He, or CH in magnetic fields of 5 to 25 × 106 gauss. Two of these have sharply defined Zeeman subcomponents, indicative of very uniform surface fields. The remaining 5 have still stronger fields, such that the spectral features if present are weak and of uncertain origin. In these objects the magnetic field is identified by the elliptical polarization of the optical continuum, and is of order 108 gauss. Within the small sample of 10 there is some evidence that the magnetic field modifies the normal extreme atm. compns. of white dwarfs. Most of the magnetic white dwarfs show no spectral or polarization variations, and may be rotating very slowly (P > 10 years). However, 2 are identified as oblique rotators with periods of the order of hours, in line with ests. of nonmagnetic white dwarfs. Other measurable properties of magnetic white dwarfs do not seem remarkably different from white dwarfs in general. The fact tha they are not esp. hot means that the time scale for field decay is comparable to or longer than that for cooling.**19**Lai, D. Matter in strong magnetic fields.*Rev. Mod. Phys.*2001,*73*, 629– 662, DOI: 10.1103/RevModPhys.73.629Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXptVyht7Y%253D&md5=47712d6d4aa8fd63065cc9b5ee72e123Matter in strong magnetic fieldsLai, DongReviews of Modern Physics (2001), 73 (3), 629-661CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. The properties of matter are drastically modified by strong magnetic fields, B»me2e3c/ℏ3 = 2.35 × 109 G (1 G = 10-4T), as are typically found on the surfaces of neutron stars. In such strong magnetic fields, the Coulomb force on an electron acts as a small perturbation compared to the magnetic force. The strong-field condition can also be mimicked in lab. semiconductors. Because of the strong magnetic confinement of electrons perpendicular to the field, atoms attain a much greater binding energy compared to the zero-field case, and various other bound states become possible, including mol. chains and 3-dimensional condensed matter. This article reviews the electronic structure of atoms, mols., and bulk matter, as well as the thermodn. properties of dense plasma, in strong magnetic fields, 109 G«B1016 G. The focus is on the basic phys. pictures and approx. scaling relations, although various theor. approaches and numerical results are also discussed. For a neutron star surface composed of light elements such as H or He, the outermost layer constitutes a nondegenerate, partially ionized Coulomb plasma if B1015 G (at temp. T 106 K), and may be as a condensed liq. if the magnetic field is stronger (and T106 K). For an Fe surface, the outermost layer of the neutron star can be in a gaseous or a condensed phase, depending on the cohesive property of the Fe condensate.**20**Ferrario, L.; de Martino, D.; Gänsicke, B. T. Magnetic White Dwarfs.*Space Sci. Rev.*2015,*191*, 111– 169, DOI: 10.1007/s11214-015-0152-0Google ScholarThere is no corresponding record for this reference.**21**Xu, S.; Jura, M.; Koester, D.; Klein, B.; Zuckerman, B. Discovery of molecular hydrogen in white dwarf atmospheres.*Astrophys. J., Lett.*2013,*766*, L18, DOI: 10.1088/2041-8205/766/2/L18Google Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXht1ajtbrP&md5=2e94bb4aade7867066121741fc8a936aDiscovery of molecular hydrogen in white dwarf atmospheresXu, S.; Jura, M.; Koester, D.; Klein, B.; Zuckerman, B.Astrophysical Journal, Letters (2013), 766 (2), L18/1-L18/3, 3 pp.CODEN: AJLEEY; ISSN:2041-8213. (IOP Publishing Ltd.)With the Cosmic Origins Spectrograph on board the Hubble Space Telescope, we have detected mol. hydrogen in the atmospheres of three white dwarfs with effective temps. below 14,000 K, G29-38, GD 133 and GD 31. This discovery provides new independent constraints on the stellar temp. and surface gravity of white dwarfs.**22**Compernolle, S.; Chibotaru, L. F.; Ceulemans, A. Vortices and Their Relation to Ring Currents and Magnetic Moments in Nanographenes in High Magnetic Field.*J. Phys. Chem. B*2006,*110*, 19340– 19351, DOI: 10.1021/jp063947hGoogle Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XptVyltrY%253D&md5=ad9a565162fd37830c9217b59311650aVortices and Their Relation to Ring Currents and Magnetic Moments in Nanographenes in High Magnetic FieldCompernolle, S.; Chibotaru, L. F.; Ceulemans, A.Journal of Physical Chemistry B (2006), 110 (39), 19340-19351CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)Much attention has been paid to the role of vortices in the magnetic response properties of superconductors, but less so for mol. systems. Here we present a theor. anal. on nanographenes subject to a strong homogeneous magnetic field. The anal. is based on the simple H.ovrddot.uckel-London model, for which we derive the topol. definition of vorticity. The results are confirmed by a more elaborate model that includes nonnearest neighbor interaction, the explicit presence of nuclei and all terms due to the magnetic field. We find that due to frontier orbital intersections, large changes in magnetic dipole moments occur. Orbital energy min. and maxima can be related to change of vortex patterns with flux.**23**Murdin, B.; Li, J.; Pang, M.; Bowyer, E.; Litvinenko, K.; Clowes, S.; Engelkamp, H.; Pidgeon, C.; Galbraith, I.; Abrosimov, N.; Riemann, H.; Pavlov, S.; Hübers, H.-W.; Murdin, P. Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf stars.*Nat. Commun.*2013,*4*, 1469, DOI: 10.1038/ncomms2466Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC3sznslKhtg%253D%253D&md5=b423aca7f3dee202fb9e1c4ec327a9c2Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf starsMurdin B N; Li Juerong; Pang M L Y; Bowyer E T; Litvinenko K L; Clowes S K; Engelkamp H; Pidgeon C R; Galbraith I; Abrosimov N V; Riemann H; Pavlov S G; Hubers H-W; Murdin P GNature communications (2013), 4 (), 1469 ISSN:.Laboratory spectroscopy of atomic hydrogen in a magnetic flux density of 10(5) T (1 gigagauss), the maximum observed on high-field magnetic white dwarfs, is impossible because practically available fields are about a thousand times less. In this regime, the cyclotron and binding energies become equal. Here we demonstrate Lyman series spectra for phosphorus impurities in silicon up to the equivalent field, which is scaled to 32.8 T by the effective mass and dielectric constant. The spectra reproduce the high-field theory for free hydrogen, with quadratic Zeeman splitting and strong mixing of spherical harmonics. They show the way for experiments on He and H(2) analogues, and for investigation of He(2), a bound molecule predicted under extreme field conditions.**24***LONDON*, A quantum chemistry program for plane-wave/GTO hybrid basis sets and finite magnetic field calculations; http://londonprogram.org (accessed December 16, 2020).Google ScholarThere is no corresponding record for this reference.**25***BAGEL, Brilliantly Advanced General Electronic-Structure Library*; http://nubakery.org (accessed December 16, 2020).Google ScholarThere is no corresponding record for this reference.**26**Williams-Young, D. B.; Petrone, A.; Sun, S.; Stetina, T. F.; Lestrange, P.; Hoyer, C. E.; Nascimento, D. R.; Koulias, L.; Wildman, A.; Kasper, J.; Goings, J. J.; Ding, F.; DePrince, A. E.; Valeev, E. F.; Li, X. The Chronus Quantum software package.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2020,*10*, e1436, DOI: 10.1002/wcms.1436Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXmvV2qsrw%253D&md5=7c0938366c14e76b901d9a54a3f4f497The Chronus Quantum software packageWilliams-Young, David B.; Petrone, Alessio; Sun, Shichao; Stetina, Torin F.; Lestrange, Patrick; Hoyer, Chad E.; Nascimento, Daniel R.; Koulias, Lauren; Wildman, Andrew; Kasper, Joseph; Goings, Joshua J.; Ding, Feizhi; DePrince, A. Eugene, III; Valeev, Edward F.; Li, XiaosongWiley Interdisciplinary Reviews: Computational Molecular Science (2020), 10 (2), e1436CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)The Chronus Quantum (ChronusQ) software package is an open source (under the GNU General Public License v2) software infrastructure which targets the soln. of challenging problems that arise in ab initio electronic structure theory. Special emphasis is placed on the consistent treatment of time dependence and spin in the electronic wave function, as well as the inclusion of relativistic effects in said treatments. In addn., ChronusQ provides support for the inclusion of uniform finite magnetic fields as external perturbations through the use of gauge-including AOs. ChronusQ is a parallel electronic structure code written in modern C++ which utilizes both message passing implementation and shared memory (OpenMP) parallelism. In addn. to the examn. of the current state of code base itself, a discussion regarding ongoing developments and developer contributions will also be provided. This article is categorized under:Software > Quantum Chem. Electronic Structure Theory > Ab Initio Electronic Structure Methods Electronic Structure Theory > D. Functional Theory.**27**Pausch, A.; Klopper, W. Efficient evaluation of three-centre two-electron integrals over London orbitals.*Mol. Phys.*2020,*118*, e1736675, DOI: 10.1080/00268976.2020.1736675Google Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXktl2gtL4%253D&md5=ec2a6c383ef9fbb479822f7406b61463Efficient evaluation of three-centre two-electron integrals over London orbitalsPausch, Ansgar; Klopper, WimMolecular Physics (2020), 118 (21-22), e1736675/1-e1736675/11CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)A review. The nonperturbative calcn. of mol. properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London AOs (LAOs). With these orbitals, the calcn. of four-center electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resoln.-of-the-identity (RI) approxn. for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approxn. is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-center integrals in the RI approxn. remain real-valued, and the three-center integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calcn. of four-center ERIs over LAOs, using the RI approxn. thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an addnl. factor of at least two. By using other well-established methods such as Cauchy-Schwarz screening, the difference-d. approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calcns. in magnetic fields can be increased even further.**28***QUEST, A rapid development platform for Quantum Electronic Structure Techniques*, 2017; quest.codes (accessed December 16, 2020).Google ScholarThere is no corresponding record for this reference.**29**London, F. Théorie quantique des courants interatomiques dans les combinaisons aromatiques.*J. Phys. Radium*1937,*8*, 397– 409, DOI: 10.1051/jphysrad:01937008010039700Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaA1cXitVahsA%253D%253D&md5=3539386fad997734786f7ad49466deaeQuantum theory of interatomic currents in aromatic compoundsLondon, F.Journal de Physique et le Radium (1937), 8 (), 397-409CODEN: JPRAAJ; ISSN:0368-3842.Math. study of the anomalous anisotropic diamagnetism observed in aromatic compds. This is explained on the basis of interat. elec. currents peculiar to these compds. Benzene, naphthalene, anthracene, biphenyl, pyrene and phenanthrene are treated as examples.**30**Reynolds, R. D.; Shiozaki, T. Fully relativistic self-consistent field under a magnetic field.*Phys. Chem. Chem. Phys.*2015,*17*, 14280– 14283, DOI: 10.1039/C4CP04027AGoogle Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhslags7bJ&md5=3e856c62040ee1ae6065b4b9f621f3e2Fully relativistic self-consistent field under a magnetic fieldReynolds, Ryan D.; Shiozaki, ToruPhysical Chemistry Chemical Physics (2015), 17 (22), 14280-14283CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We present a gauge-invariant implementation of the four-component Dirac-Hartree-Fock method for simulating the electronic structure of heavy element complexes in magnetic fields. The addnl. cost assocd. with the magnetic field is shown to be only 10-13% of that at zero field. The Dirac-Hartree-Fock wave function is constructed from gauge-including AOs. The so-called restricted magnetic balance is used to generate 2-spinor basis functions for the small component. The mol. integrals for the Coulomb and Gaunt interactions are computed using d. fitting. Our efficient, parallel implementation allows for simulating the electronic structure of mols. contg. more than 100 atoms with a few heavy elements under magnetic fields.**31**Schlegel, H. B.; Frisch, M. J. Transformation between Cartesian and pure spherical harmonic Gaussians.*Int. J. Quantum Chem.*1995,*54*, 83– 87, DOI: 10.1002/qua.560540202Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXktlOmtLY%253D&md5=2971b5414f3b3b740f33a15b756323abTransformation between Cartesian and pure spherical harmonic GaussiansSchlegel, H. Bernhard; Freisch, Michael J.International Journal of Quantum Chemistry (1995), 54 (2), 83-7CODEN: IJQCB2; ISSN:0020-7608. (Wiley)Spherical Gaussians can be expressed as linear combinations of the appropriate Cartesian Gaussians. General expressions for the transformation coeffs. are given. Values for the transformation coeffs. are tabulated up to h-type functions.**32**Helgaker, T.; Jørgensen, P.; Olsen, J.*Molecular Electronic-Structure Theory*; John Wiley & Sons, 2000; DOI: 10.1002/9781119019572 .Google ScholarThere is no corresponding record for this reference.**33**McMurchie, L.; Davidson, E. One- and two-electron integrals over Cartesian Gaussian functions.*J. Comput. Phys.*1978,*26*, 218– 231, DOI: 10.1016/0021-9991(78)90092-XGoogle Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE1cXhsFyksb4%253D&md5=119248f25c59392ff1d9ce1c7285b3c7One- and two-electron integrals over Cartesian Gaussian functionsMcMurchie, Larry E.; Davidson, Ernest R.Journal of Computational Physics (1978), 26 (2), 218-31CODEN: JCTPAH; ISSN:0021-9991.A formalism was developed that allows overlap, kinetic-energy, potential-energy, and electron-repulsion integrals over Cartesian Gaussian wave functions to be expressed in very compact forms involving easily calcd. auxiliary functions. Similar formulas involving the same auxiliary functions are given for the common charge moments, elec.-field operators, and spin-interaction operators. Recursion relations are given for the auxiliary functions, which make it possible to use Gaussian wave functions having arbitrarily large angular momentum. An algorithm is given for calcg. electron-repulsion integrals.**34**McMurchie, L. E.; Davidson, E. R. One- and two-electron integrals over Cartesian Gaussian functions.*J. Comput. Phys.*1978,*26*, 218– 231, DOI: 10.1016/0021-9991(78)90092-XGoogle Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE1cXhsFyksb4%253D&md5=119248f25c59392ff1d9ce1c7285b3c7One- and two-electron integrals over Cartesian Gaussian functionsMcMurchie, Larry E.; Davidson, Ernest R.Journal of Computational Physics (1978), 26 (2), 218-31CODEN: JCTPAH; ISSN:0021-9991.**35**Fortunelli, A.; Salvetti, O. Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functions.*Int. J. Quantum Chem.*1993,*48*, 257– 265, DOI: 10.1002/qua.560480407Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXhsFejsA%253D%253D&md5=32a6b01291a224985215ef9ec0097335Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functionsFortunelli, Alessandro; Salvetti, OrianoInternational Journal of Quantum Chemistry (1993), 48 (4), 257-65CODEN: IJQCB2; ISSN:0020-7608.Recurrence relations are derived for the evaluation of two-electron repulsion integrals (ERIs) over Hermite and spherical Gaussian functions. Through such relations, a generic ERI or ERI deriv. may be reduced to "basic" integrals, i.e., true and auxiliary integrals involving only zero angular momentum functions.**36**Reine, S.; Tellgren, E.; Helgaker, T. A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians.*Phys. Chem. Chem. Phys.*2007,*9*, 4771, DOI: 10.1039/b705594cGoogle Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXptlSqsrc%253D&md5=04c2f3ae05a68ce9d39cc8909765c3d8A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic GaussiansReine, Simen; Tellgren, Erik; Helgaker, TrygvePhysical Chemistry Chemical Physics (2007), 9 (34), 4771-4779CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Utilizing the fact that solid-harmonic combinations of Cartesian and Hermite Gaussian AOs are identical, a new scheme for the evaluation of mol. integrals over solid-harmonic AOs is presented, where the integration is carried out over Hermite rather than Cartesian AOs. Since Hermite Gaussians are defined as derivs. of spherical Gaussians, the corresponding mol. integrals become the derivs. of integrals over spherical Gaussians, whose transformation to the solid-harmonic basis is performed in the same manner as for integrals over Cartesian Gaussians, using the same expansion coeffs. The presented solid-harmonic Hermite scheme simplifies the evaluation of deriv. mol. integrals, since differentiation by nuclear coordinates merely increments the Hermite quantum nos., thereby providing a unified scheme for undifferentiated and differentiated four-center mol. integrals. For two- and three-center two-electron integrals, the solid-harmonic Hermite scheme is particularly efficient, significantly reducing the cost relative to the Cartesian scheme.**37**Colle, R.; Fortunelli, A.; Simonucci, S. A mixed basis set of plane waves and Hermite-Gaussian functions. Analytic expressions of prototype integrals.*Nuovo Cimento Soc. Ital. Fis., D*1987,*9*, 969– 977, DOI: 10.1007/BF02464849Google ScholarThere is no corresponding record for this reference.**38**Colle, R.; Fortunelli, A.; Simonucci, S. Hermite-Gaussian functions modulated by plane waves: a general basis set for bound and continuum states.*Nuovo Cimento Soc. Ital. Fis., D*1988,*10*, 805– 818, DOI: 10.1007/BF02450141Google ScholarThere is no corresponding record for this reference.**39**Tachikawa, M.; Shiga, M. Evaluation of atomic integrals for hybrid Gaussian type and plane-wave basis functions via the McMurchie-Davidson recursion formula.*Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top.*2001,*64*, 056706, DOI: 10.1103/PhysRevE.64.056706Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXotVKgur0%253D&md5=4765ba25da04adee7437836b6d14b0dfEvaluation of atomic integrals for hybrid Gaussian type and plane-wave basis functions via the McMurchie-Davidson recursion formulaTachikawa, Masanori; Shiga, MotoyukiPhysical Review E: Statistical, Nonlinear, and Soft Matter Physics (2001), 64 (5-2), 056706/1-056706/4CODEN: PRESCM ISSN:. (American Physical Society)A convenient formalism is developed for the evaluation of at. integrals composed of a hybrid Gaussian type function and plane-wave (GTF-PW) basis set, based upon the recursion scheme proposed by McMurchie and Davidson [L. E. McMurchie and E. R. Davidson, J. Comput. Phys. 26, 218 (1978)] which was originally for Gaussian type basis functions. We show that revisions of recursion relations in the original article are necessary in order to allow systematic prodn. of overlap, kinetic energy, nuclear attraction, and electron repulsion integrals in compact forms. Involving easy calcn. of complex incomplete gamma functions, the recursion relations enable the use of hybrid GTF-PW basis functions with arbitrarily large angular momentum. This basis function can be applied to the first-principle calcn. for solids involving localized electron orbitals.**40**Kanno, M.; Kato, T.; Kono, H.; Fujimura, Y.; Faisal, F. H. M. Incorporation of a wave-packet propagation method into the S-matrix framework: Investigation of the effects of excited state dynamics on intense-field ionization.*Phys. Rev. A: At., Mol., Opt. Phys.*2005,*72*, 033418, DOI: 10.1103/PhysRevA.72.033418Google Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXhtFCmsbnF&md5=dbefcfb046e6d470fd48147f59bf3944Incorporation of a wave-packet propagation method into the S-matrix framework: Investigation of the effects of excited state dynamics on intense-field ionizationKanno, Manabu; Kato, Tsuyoshi; Kono, Hirohiko; Fujimura, Yuichi; Faisal, Farhad H. M.Physical Review A: Atomic, Molecular, and Optical Physics (2005), 72 (3, Pt. B), 033418/1-033418/14CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The authors propose a theor. method for study of ionization of atoms and mols. in intense laser fields that copes with the effects of excited state dynamics (or intramol. electronic dynamics). The time-evolving wave packet composed of only bound electronic states, |Φi(t)〉, is introduced into a framework of the intense-field S-matrix theory. Then, the effects of both Coulomb field and radiation field on the bound electron(s) are well described by |Φi(t)〉, while the effects of a radiation field on a freed electron are also treated in a nonperturbative way. The authors have applied the theory to ionization of H and H2+ in ultrashort intense laser pulses. Although only a small no. of Gaussian functions were used in the expansion of |Φi(t)〉, the present method can quant. reproduce the features of enhanced ionization of H2+ obtained by an accurate grid propagation method. This agreement supports the view that field-induced population transfer between the lowest two electronic states triggers the enhancement of ionization at large internuclear distances. The authors also applied the method to calc. the photoelectron momentum distribution of H in an intense near-IR field. A broad low intensity component due to rescattering appears in the distribution of the momentum perpendicular to the polarization direction of an applied laser field, as obsd. in the expts. of single ionization of noble gas atoms. The present method provides a practical way of properly describing the nonperturbative nature of field-induced dynamics of an electron (or electrons) in the presence of both Coulomb and radiation fields.**41**Obara, S.; Saika, A. Efficient recursive computation of molecular integrals over Cartesian Gaussian functions.*J. Chem. Phys.*1986,*84*, 3963, DOI: 10.1063/1.450106Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL28XitFeguro%253D&md5=7643c24c26bd4ea9e37424fb8cf66935Efficient recursive computation of molecular integrals over Cartesian Gaussian functionsObara, S.; Saika, A.Journal of Chemical Physics (1986), 84 (7), 3963-74CODEN: JCPSA6; ISSN:0021-9606.Recurrence expressions for calcg. various types of mol. integrals over Cartesian Gaussian functions were derived by using the recurrence formula for three-center overlap integrals. A no. of characteristics inherent in the recursive formalism allowed an efficient algorithm to be developed for mol.-integral computations. With respect to electron-repulsion integrals and their derivs., the present algorithm, with a significant saving of computer time, was superior to other currently available methods. A long innermost loop incorporated in the present scheme facilitates a fast computation on a vector-processing computer.**42**Obara, S.; Saika, A. General recurrence formulas for molecular integrals over Cartesian Gaussian functions.*J. Chem. Phys.*1988,*89*, 1540, DOI: 10.1063/1.455717Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXltlGitbc%253D&md5=7598cac04c1b8129a3084cb66b237930General recurrence formulas for molecular integrals over Cartesian Gaussian functionsObara, S.; Saika, A.Journal of Chemical Physics (1988), 89 (3), 1540-59CODEN: JCPSA6; ISSN:0021-9606.General recurrence formulas for various types of one- and two-electron mol. integrals over Cartesian Gaussian functions are derived by introducing basic integrals. These formulas are capable of dealing with (1) mol. integrals with any spatial operators in the nonrelativistic forms of the relativistic wave equations; (2) those with the kernel of the Fourier transform; (3) those with arbitrarily defined spatial operators so far as the integrals can be expressed in terms of the basic integrals; and (4) any order of their derivs. with respect to the function centers in the above integrals. Thus, the present formulation can cover a large class of mol. integrals necessary for theor. studies of mol. systems by ab initio calcns., and furthermore provides us with an efficient scheme of computing them by virtue of its recursive nature.**43**Shavitt, I.*Methods in Computational Physics*; Academic Press: New York, 1963; Vol. 3; pp 1– 45.Google ScholarThere is no corresponding record for this reference.**44**Saunders, V. R.*Computational Techniques in Quantum Chemistry and Molecular Physics*; Springer: Dordrecht, The Netherlands, 1975; pp 347– 424.Google ScholarThere is no corresponding record for this reference.**45**Gill, P. M. W.*Adv. Quantum Chem.*; Elsevier: BV, 1994; pp 141– 205.Google ScholarThere is no corresponding record for this reference.**46**Boys, S. F. Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System.*Proc. R. Soc. A*1950,*200*, 542– 554, DOI: 10.1098/rspa.1950.0036Google Scholar46https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaG3cXktVShtA%253D%253D&md5=1bf132315d97130502916898345e9b31Electronic wave functions. I. A general method of calculation for the stationary states of any molecular systemBoys, S. F.Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1950), 200 (), 542-54CODEN: PRLAAZ; ISSN:1364-5021.By taking Gaussian functions, and functions derived from these by differentiation with respect to the parameters, complete systems of functions can be constructed appropriate to any mol. problem, and all the necessary integrals can be explicitly evaluated. The only obstacle to the evaluation of wave functions of any required degree of accuracy is the labor of computation. The methods developed give for the first time a quant. method of evaluating the stationary-state wave functions and energy levels of all atoms and mols. to any required degree of accuracy.**47**Čársky, P.; Polášek, M. Incomplete Gamma Fm(x) Functions for Real Negative and Complex Arguments.*J. Comput. Phys.*1998,*143*, 259– 265, DOI: 10.1006/jcph.1998.5975Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXkt1Oiu7o%253D&md5=7a63cbbd4b75f93901ec8b20450c8f01Incomplete gamma Fm(x) functions for real negative and complex argumentsCarsky, Petr; Polasek, MartinJournal of Computational Physics (1998), 143 (1), 259-265CODEN: JCTPAH; ISSN:0021-9991. (Academic Press)Incomplete gamma functions Fm(x), originally defined and used in the electronic structure theory, have been examd. from the viewpoint of electron-mol. scattering theory for their possible use in calcn. of two-electron integrals in a mixed Gaussian and plane-wave basis set. Effective calcn. of Fm(z) functions with a complex argument is discussed. (c) 1998 Academic Press.**48**Ishida, K. Accurate and fast algorithm of the molecular incomplete gamma function with a complex argument.*J. Comput. Chem.*2004,*25*, 739– 748, DOI: 10.1002/jcc.20002Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXis1Kktb4%253D&md5=baa4c691923a9780dc754aad86097e1aAccurate and fast algorithm of the molecular incomplete gamma function with a complex argumentIshida, KazuhiroJournal of Computational Chemistry (2004), 25 (5), 739-748CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)Several efficient algorithms for the accurate and fast calcn. of the mol. incomplete gamma function Fm(z) with a complex argument z are developed. The complex incomplete gamma function is arising in mol. integrals over the gauge-including AOs. Two kinds of algorithms are recommended: (1) a high-precision version and (2) a fast version. The high-precision version is able to guarantee 15 significant figures (10-15 in the relative error) and the fast version is able to guarantee 12 significant figures (10-12 in the relative error), at worst, within the double-precision arithmetic. The fast version is about 5-20 times faster than the high-precision version. For most mol. calcns., the fast version will give a satisfied precision.**49**Mathar, R. J. Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument.*Numer. Algorithms*2004,*36*, 247– 264, DOI: 10.1023/B:NUMA.0000040063.91709.58Google ScholarThere is no corresponding record for this reference.**50**Helgaker, T.; Taylor, P. R. On the evaluation of derivatives of Gaussian integrals.*Theor. Chim. Acta*1992,*83*, 177– 183, DOI: 10.1007/BF01132826Google ScholarThere is no corresponding record for this reference.**51**King, H. F.; Dupuis, M. Numerical integration using Rys polynomials.*J. Comput. Phys.*1976,*21*, 144– 165, DOI: 10.1016/0021-9991(76)90008-5Google ScholarThere is no corresponding record for this reference.**52**Dupuis, M.; Rys, J.; King, H. F. Evaluation of molecular integrals over Gaussian basis functions.*J. Chem. Phys.*1976,*65*, 111– 116, DOI: 10.1063/1.432807Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE28XkvF2nsb4%253D&md5=1ede1bceba64efb430a3a383443c55ceEvaluation of molecular integrals over Gaussian basis functionsDupuis, Michel; Rys, John; King, Harry F.Journal of Chemical Physics (1976), 65 (1), 111-16CODEN: JCPSA6; ISSN:0021-9606.The efficient computation of the ubiquitous electron repulsion integral in mol. quantum mechanics was studied. Differences and similarities in organization of existing Gaussian integral programs are discussed, and a new strategy is developed. An anal. based on the theory of orthogonal polynomials yields a general formula for basis functions of arbitrarily high angular momentum. This method is simple, numerically well behaved, and was incorporated into a new mol. SCF program HONDO. Preliminary tests indicate that it is competitive with existing methods esp. for highly angularly dependent functions.**53**Rys, J.; Dupuis, M.; King, H. F. Computation of electron repulsion integrals using the Rys quadrature method.*J. Comput. Chem.*1983,*4*, 154– 157, DOI: 10.1002/jcc.540040206Google Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXit1Wqu7s%253D&md5=f8d3d4f8d5c3f1fc2492554f38b88bdcComputation of electron repulsion integrals using the Rys quadrature methodRys, J.; Dupuis, M.; King, H. F.Journal of Computational Chemistry (1983), 4 (2), 154-7CODEN: JCCHDD; ISSN:0192-8651.Following an earlier proposal to evaluate electron repulsion integrals over Gaussian basis functions by a numerical quadrature based on a set of orthogonal polynomials (Rys polynomials), a computational procedure is outlined for efficient evaluation of the two-dimensional integrals. Compact recurrence formulas for the integrals make the method particularly fitted to handle high-angular-momentum basis functions.**54**Čársky, P.; Polášek, M. Evaluation of Molecular Integrals in a Mixed Gaussian and Plane-Wave Basis by Rys Quadrature.*J. Comput. Phys.*1998,*143*, 266– 277, DOI: 10.1006/jcph.1998.5976Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXkt1Oiu7s%253D&md5=68620231706ab0aae7fed92b069be57eEvaluation of molecular integrals in a mixed Gaussian and plane-wave basis by Rys quadratureCarsky, Petr; Polasek, MartinJournal of Computational Physics (1998), 143 (1), 266-277CODEN: JCTPAH; ISSN:0021-9991. (Academic Press)We report on the use of Rys numerical quadrature for the calcn. of two-electron exchange integrals contg. two Gaussians and two plane-wave functions, and two-electron integrals contg. three Gaussians and one plane-wave function. Generally, the Rys polynomials for this mixed basis set are complex. We present formulas for obtaining their roots and wts. that are also generally complex. Rys numerical quadrature provides an alternative method for calcn. of integrals of this type that are encountered in the electron-mol. scattering theory. (c) 1998 Academic Press.**55**Head-Gordon, M.; Pople, J. A. A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relations.*J. Chem. Phys.*1988,*89*, 5777, DOI: 10.1063/1.455553Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXivFajuw%253D%253D&md5=c986058fd86ea827e91bb3ea2ac57519A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relationsHead-Gordon, Martin; Pople, John A.Journal of Chemical Physics (1988), 89 (9), 5777-86CODEN: JCPSA6; ISSN:0021-9606.An efficient method is presented for evaluating two-electron Cartesian Gaussian integrals, and their first derivs. with respect to nuclear coordinates. It is based on the recurrence relation (RR) of Obara and Saika (1986), and an addnl. new RR, which are combined together in a general algorithm applicable to any angular momenta. This algorithm exploits the fact that the new RR can be applied outside contraction loops. By floating point operation counts and comparative timings it is shown to be generally superior to existing methods, particularly for basis sets contg. d functions.**56**Lindh, R.; Ryu, U.; Liu, B. The reduced multiplication scheme of the Rys quadrature and new recurrence relations for auxiliary function based two-electron integral evaluation.*J. Chem. Phys.*1991,*95*, 5889, DOI: 10.1063/1.461610Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXmtl2qtLo%253D&md5=0915bf9171b22ea14f8a64e9df982b58The reduced multiplication scheme of the Rys quadrature and new recurrence relations for auxiliary function based two-electron integral evaluationLindh, R.; Ryu, U.; Liu, B.Journal of Chemical Physics (1991), 95 (8), 5889-97CODEN: JCPSA6; ISSN:0021-9606.A reduced-multiplication algorithm for the Rys quadrature is presented. The method is based on new ways in which the Rys quadrature can be developed if it is implemented together with the transfer equation applied to the contracted integrals. In parallel to the new algorithm for the Rys quadrature, improvements are suggested to the auxiliary-function-based algorithms. The two new methods have very favorable theor. floating point operation (FLOP) counts as compared to other methods. It is noted that the only significant difference in performance of the two new methods is due to the vectorizability of the presented algorithms. In order to exhibit this, both methods were implemented in the integral program SEWARD. Timings are presented for comparisons with other implementations. Finally, it is demonstrated how the transfer equation in connection with the use of spherical harmonic Gaussians offers a very attractive path to compute the two-electron integrals of such basis functions. It is demonstrated both theor. and with actual performance that the use of spherical harmonic Gaussians offers a clear advantage over the traditional evaluation of the two-electron integrals in the Cartesian Gaussian basis.**57**Lindh, R. The reduced multiplication scheme of the Rys-Gauss quadrature for 1st order integral derivatives.*Theor. Chim. Acta*1993,*85*, 423– 440, DOI: 10.1007/BF01112982Google Scholar57https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXls1Oqur0%253D&md5=f433efff3d647af6c17351fee6d70a31The reduced multiplication scheme of the Rys-Gauss quadrature for 1st order integral derivativesLindh, RolandTheoretica Chimica Acta (1993), 85 (6), 423-40CODEN: TCHAAM; ISSN:0040-5744.An implementation of the reduced multiplication scheme of the Rys-Gauss quadrature to compute the gradients of electron repulsion integrals is discussed. The Rys-Gauss quadrature is very suitable for efficient utilization of simplifications as offered by the direct computation of symmetry adapted gradients and the use of the translational invariance of the integrals. The introduction of the so-called intermediate products is also demonstrated to further reduce the floating point operation count. Two prescreening techniques based on the 2nd order d. matrix in the basis of the uncontracted Gaussian functions is proposed and investigated in this paper. It is not necessary to employ the Cauchy-Schwarz inequality to achieve efficient prescreening. All the features mentioned above were demonstrated by their implementation into the gradient program ALASKA. The paper offers a theor. and practical assessment of the modified Rys-Gauss quadrature in comparison with other methods and implementations and a detailed anal. of the behavior of the method as suggested above as a function of changes with respect to symmetry, basis set quality, mol. size, and prescreening threshold.**58**Komornicki, A.; Ishida, K.; Morokuma, K.; Ditchfield, R.; Conrad, M. Efficient determination and characterization of transition states using ab-initio methods.*Chem. Phys. Lett.*1977,*45*, 595– 602, DOI: 10.1016/0009-2614(77)80099-7Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2sXht1Omsrw%253D&md5=9f2041a5838eb8d70a4653813bf06813Efficient determination and characterization of transition states using ab-initio methodsKomornicki, Andrew; Ishida, Kazuhiro; Morokuma, Keiji; Ditchfield, Robert; Conrad, MorganChemical Physics Letters (1977), 45 (3), 595-602CODEN: CHPLBC; ISSN:0009-2614.The gradient of the potential energy with respect to the nuclear coordinates was calcd. by using ab-initio single-determinant MO methods. The calcd. gradient was used together with very efficient minimization methods to locate and characterize transition states on many-dimensional potential-energy surfaces. Previously such methods have only been applied to semiempirical potential functions. Although the calcn. of the gradient in addn. to the energy increases the computational time by about a factor of four, the feasibility of these calcns. was demonstrated by locating the transition state for the model rearrangement of HNC to HCN by using both minimal (STO-3G) and split-valence-shell (4-31G) basis sets. Further use of such methods is discussed in the direct application of ab-initio wave functions to dynamical investigations.**59**Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas.*Phys. Rev.*1964,*136*, B864– B871, DOI: 10.1103/PhysRev.136.B864Google ScholarThere is no corresponding record for this reference.**60**Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects.*Phys. Rev.*1965,*140*, A1133– A1138, DOI: 10.1103/PhysRev.140.A1133Google ScholarThere is no corresponding record for this reference.**61**Vignale, G.; Rasolt, M. Density-functional theory in strong magnetic fields.*Phys. Rev. Lett.*1987,*59*, 2360– 2363, DOI: 10.1103/PhysRevLett.59.2360Google Scholar61https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXislSqtA%253D%253D&md5=6c0439dbbbddf191b1da1f73d69ed733Density-functional theory in strong magnetic fieldsVignale, G.; Rasolt, MarkPhysical Review Letters (1987), 59 (20), 2360-3CODEN: PRLTAO; ISSN:0031-9007.The current-d.-functional theory is formulated for systems in arbitrarily strong magnetic fields. A set of self-consistent equations comparable to the Kohn-Sham equations for ordinary d.-functional theory is derived, and proved to be gage invariant and to satisfy the continuity equation. The exchange-correlation energy functional Exc[n,jp] [n(r) is the d. and jp(r) is the "paramagnetic" c.d.] depends on the current via the combination v(r) = V × [jp(r)/n(r)]. An explicit formula for Exc is derived, which is local in v(r).**62**Vignale, G.; Rasolt, M. Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields.*Phys. Rev. B: Condens. Matter Mater. Phys.*1988,*37*, 10685– 10696, DOI: 10.1103/PhysRevB.37.10685Google Scholar62https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2sfhtFaqug%253D%253D&md5=08fd751039a075c8cc57ec29f67246ffCurrent- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fieldsVignale; RasoltPhysical review. B, Condensed matter (1988), 37 (18), 10685-10696 ISSN:0163-1829.There is no expanded citation for this reference.**63**Dobson, J. F. Interpretation of the Fermi hole curvature.*J. Chem. Phys.*1991,*94*, 4328– 4333, DOI: 10.1063/1.460619Google Scholar63https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXitFCrsLs%253D&md5=bd726fc23d10d626f433e386fbd26816Interpretation of the Fermi hole curvatureDobson, John F.Journal of Chemical Physics (1991), 94 (6), 4328-33CODEN: JCPSA6; ISSN:0021-9606.Two different interpretations are given for the Fermi-hole-curvature parameter in many-electron systems used recently to est. the size of the correlation hole, to clarify aspects of chem. shell structure and bonding, and to discuss mobility of the Fermi hole. The first, more straightforward interpretation involves the no. of "other" electrons to be found in a small neighborhood near a given electron. The notion of other electrons leads naturally to correlation functionals, which correctly vanish when only one electron is present. The second interpretation, made explicit by use of the Wigner pair distribution, involves the d. of relative kinetic energy of pairs of spin-parallel electrons at point r. Since, in a classical interpretation at least, the correlation hole in a nonuniform Coulomb system depends both on d. and relative kinetic energy of colliding pairs, one expects that both the Fermi hole curvature and the d. should be significant in constructing theories of the correlation energy of such systems.**64**Dobson, J. F. Alternative expressions for the Fermi hole curvature.*J. Chem. Phys.*1993,*98*, 8870– 8872, DOI: 10.1063/1.464444Google Scholar64https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXkvVSmt74%253D&md5=0623ae49360b47df1b9560a86baffe77Alternative expressions for the Fermi hole curvatureDobson, John F.Journal of Chemical Physics (1993), 98 (11), 8870-2CODEN: JCPSA6; ISSN:0021-9606.The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero sepn. r' = r for a pair of fermions in a many-fermion system. It has been used by a no. of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-fermion state and is not limited to the single-determinant case investigated previously.**65**Becke, A. D. Current-density dependent exchange-correlation functionals.*Can. J. Chem.*1996,*74*, 995– 997, DOI: 10.1139/v96-110Google Scholar65https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xks1yrsbY%253D&md5=4198a87b2a632236782239cbfafce319Current-density dependent exchange-correlation functionalsBecke, Axel D.Canadian Journal of Chemistry (1996), 74 (6), 995-997CODEN: CJCHAG; ISSN:0008-4042. (National Research Council of Canada)Previous models for exchange (Becke and Roussel, Phys. Rev. A:39, 3761 (1989)) and for correlation (Becke, J. Chem. Phys. 88, 1053 (1988)) are, in a simple and natural way, generalized to include explicit dependence on c.d. J. First-principles incorporation of J into exchange-correlation d. functionals, as proposed here, is crucial for further progress in the study of magnetic effects in d.-functional theory.**66**Becke, A. D. Current density in exchange-correlation functionals: Application to atomic states.*J. Chem. Phys.*2002,*117*, 6935– 6938, DOI: 10.1063/1.1503772Google Scholar66https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XnsF2itro%253D&md5=37fd49dbccbeadc7d61935a97183f1cfCurrent density in exchange-correlation functionals: Application to atomic statesBecke, Axel D.Journal of Chemical Physics (2002), 117 (15), 6935-6938CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An old and yet unsolved problem in d.-functional theory is the strong dependence of degenerate open-shell at. energies on the occupancy of the AOs. This arises from the fact that degenerate AOs of different ml do not have equiv. densities. Approx. d. functionals therefore give energies depending strongly on which orbitals are occupied. This problem is solved in the present work by incorporating c.d. into the calcns. using a current-d. dependent functional previously published by the author.**67**Neumann, R.; Nobes, R. H.; Handy, N. C. Exchange functionals and potentials.*Mol. Phys.*1996,*87*, 1– 36, DOI: 10.1080/00268979600100011Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XhvFegs7c%253D&md5=14030dab7e55c7c6faaf9737dbcbcb37Exchange functionals and potentialsNeumann, Ralf; Nobes, Ross H.; Handy, Nicholas C.Molecular Physics (1996), 87 (1), 1-36CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis)The commonly used exchange-correlation functionals of d. functional theory and their potentials are examd. numerically following the first such investigation of Perdew. They are also investigated for Ne and Kr. Their behavior for large gradients of the d. and for large distances is not satisfactory. In particular, the correct asymptotic r-1 behavior is difficult to achieve. Following van Leeuwen and Baerends, this is linked to the energy εmax of the highest occupied orbital arising from the Kohn-Sham equations. This deficiency is linked also with the poor prediction of mol. polarizabilities. The Becke-Roussel (BR) exchange functional is examd., which is derived by assuming a hydrogen-like exchange hole at all spatial points, and it has the attraction of being dependent on both the kinetic energy d. and the Laplacian of the d. and has no adjustable parameters. Becke has presented encouraging results using this functional in a hybrid manner. Fully self-consistent Kohn-sham calcns. are performed using it in combination with Perdew's 1986 correlation functional. The results are very encouraging indeed, so much so that this exchange functional is the best generalized gradient approxn. (GGA) yet discovered. In particular, bond lengths of many mol. show a substantial improvement over results from other GGAs. For example, many CH bonds are now within exptl. accuracy, instead of being typically 0.02 Å too long. Our ab initio understanding of non-dynamic correlation and dynamic correlation is then linked with d. functional theory. It is argued that correlation functionals should pick up the local dynamic correlation, whereas exchange functionals should include non-dynamic correlation effects. For these reasons it is considered that exchange functionals are best modeled on a system for which there is effectively no non-dynamic correlation, for which the optimum example is the Ne atom. Thus, again following Becke and Roussel, the spherically averaged Hartree-Fock exchange hole for Ne is examd., compared with the BR model functional hole. An excellent overlap is found, and thus the above good results are explained. As a final contribution, the dissocn. of the H2 mol. is re-examd., looking at it in terms of the exchange hole. For a ref. electron near one proton A, the RHF model has half an exchange electron near it, and half the exchange electron near the other proton B, whereas the BR functional has one electron near the other proton b, whereas the Br functional has one electron near A, which is the correct picture. For this reason the (restricted) BR functional gives a greatly improved dissocn. curve for H2 when compared with the Hartree-Fock curve. In summary, the Becke-Roussel functional is found to be a most attractive exchange functional.**68**Pople, J. A.; Gill, P. M.; Johnson, B. G. Kohn-Sham density-functional theory within a finite basis set.*Chem. Phys. Lett.*1992,*199*, 557– 560, DOI: 10.1016/0009-2614(92)85009-YGoogle Scholar68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXhtV2qsrg%253D&md5=8fe4c7a7e9c3f9f7282d3234d343b03cKohn-Sham density-functional theory within a finite basis setPople, John A.; Gill, Peter M. W.; Johnson, Benny G.Chemical Physics Letters (1992), 199 (6), 557-60CODEN: CHPLBC; ISSN:0009-2614.The Kohn-Sham self-consistent equations, using a finite orbital basis expansion, are formulated for exchange-correlation functionals which depend on local densities and their gradients. It is shown that these can be solved iteratively without evaluation of d. Hessians. A general expression is given for the energy gradient (with respect to nuclear motion) after self-consistency has been achieved.**69**Schmidt, G. D.; Liebert, J.; Harris, H. C.; Dahn, C. C.; Leggett, S. K. Discovery of a Highly Magnetic White Dwarf with Strong Carbon Features.*Astrophys. J.*1999,*512*, 916– 919, DOI: 10.1086/306819Google Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXislaqtrk%253D&md5=1d96e31cfa7b0f8fa1c410908163ac45Discovery of a highly magnetic white dwarf with strong carbon featuresSchmidt, Gary D.; Liebert, James; Harris, Hugh C.; Dahn, Conard C.; Leggett, S. K.Astrophysical Journal (1999), 512 (2, Pt. 1), 916-919CODEN: ASJOAB; ISSN:0004-637X. (University of Chicago Press)Systematic follow-up of high proper-motion stars has identified a new cool magnetic white dwarf that displays spectacular absorption bands in the range 4200-6500 Å. The spectrum bears a striking resemblance to that of LP 790-29, a magnetic DQ star dominated by what are apparently Zeeman-shifted Swan bands of C2. However, key differences in the detailed spectra, polarization, and temp. of the two stars indicate that instead LHS 2229 may represent the 1st case of a magnetic peculiar DQ white dwarf, where absorption in the optical is produced by C2H or another C-H compd. Crude arguments suggest that the field strength on LHS 2229 is in the neighborhood of 108 G. B VI photometry proves to be effective in identifying such peculiar stars, since they lie well outside the main white dwarf sequence in a color-color diagram.**70**Reid, I. N.; Liebert, J.; Schmidt, G. D. Discovery of a Magnetic DZ White Dwarf with Zeeman-Split Lines of Heavy Elements.*Astrophys. J.*2001,*550*, L61– L63, DOI: 10.1086/319481Google Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXjsFSjsLY%253D&md5=0c7d19310fec0a907b2ad093e7596634Discovery of a magnetic DZ white dwarf with Zeeman-split lines of heavy elementsReid, I. Neill; Liebert, James; Schmidt, Gary D.Astrophysical Journal (2001), 550 (1, Pt. 2), L61-L63CODEN: ASJOAB; ISSN:0004-637X. (University of Chicago Press)A spectroscopic survey of unstudied Luyten half-second proper-motion stars has resulted in the discovery of an unusual new magnetic white dwarf. LHS 2534 proves to be the first magnetic DZ, showing Zeeman-split Na I and Mg I components, as well as Ca I and Ca II lines for which Zeeman components are blended. The Na I splittings result in a mean surface field strength est. of 1.92 MG. Apart from the magnetic field, LHS 2534 is one of the most heavily blanketed and coolest DZ white dwarfs at Teff ∼ 6000 K.**71**Kawka, A.; Vennes, S.; Ferrario, L.; Paunzen, E. Evidence of enhanced magnetism in cool, polluted white dwarfs.*Mon. Not. R. Astron. Soc.*2019,*482*, 5201– 5210, DOI: 10.1093/mnras/sty3048Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVeltL8%253D&md5=cfe4c129efc8afb3098cc16cb10fb1b9Evidence of enhanced magnetism in cool, polluted white dwarfsKawka, Adela; Vennes, Stephane; Ferrario, Lilia; Paunzen, ErnstMonthly Notices of the Royal Astronomical Society (2019), 482 (4), 5201-5210CODEN: MNRAA4; ISSN:1365-2966. (Oxford University Press)We report the discovery of a new, polluted, magnetic white dwarf in the Luyten survey of high-proper motion stars. High-dispersion spectra of NLTT 7547 reveal a complex heavy element line spectrum in a cool (≈5200 K) hydrogen-dominated atm. showing the effect of a surface averaged field of 163 kG, consistent with a 240 kG centered dipole, although the actual field structure remains uncertain. The abundance pattern shows the effect of accreted material with a distinct magnesium-rich flavor. Combined with earlier identifications, this discovery supports a correlation between the incidence of magnetism in cool white dwarfs and their contamination by heavy elements.**72**Jones, M. D.; Ortiz, G.; Ceperley, D. M. Hartree-Fock studies of atoms in strong magnetic fields.*Phys. Rev. A: At., Mol., Opt. Phys.*1996,*54*, 219– 231, DOI: 10.1103/PhysRevA.54.219Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xkt1OktLc%253D&md5=f071495632dea366d6658f94fbf7c2baHartree-Fock studies of atoms in strong magnetic fieldsJones, Matthew D.; Ortiz, Gerardo; Ceperley, David M.Physical Review A: Atomic, Molecular, and Optical Physics (1996), 54 (1), 219-231CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We present comprehensive calcns. of the electronic structure of selected first-row atoms in uniform magnetic fields of strength ≤1010 G, within a flexible implementation of the Hartree-Fock formalism. Ground-state and low-lying excited-state properties are presented for first-row atoms He, Li, C, and ion H-. We predict and describe a series of ground-state quantum transitions as a function of magnetic field strength. Due to its astrophys. importance, highly excited states of neutral He are also computed. Comparisons are made with previous works, where available.**73**Ivanov, M. V.; Schmelcher, P. Ground state of the carbon atom in strong magnetic fields.*Phys. Rev. A: At., Mol., Opt. Phys.*1999,*60*, 3558– 3568, DOI: 10.1103/PhysRevA.60.3558Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXmvFKlsLs%253D&md5=71b04a3f6753da21a506c3b6d9b0ed3bGround state of the carbon atom in strong magnetic fieldsIvanov, M. V.; Schmelcher, P.Physical Review A: Atomic, Molecular, and Optical Physics (1999), 60 (5), 3558-3568CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The ground and a few excited states of the carbon atom in external uniform magnetic fields are calcd. by means of our two-dimensional mesh Hartree-Fock method for field strengths ranging from zero up to 2.35 × 109 T. With increasing field strength the ground state undergoes six transitions involving seven different electronic configurations which belong to three groups with different spin projections Sz = -1, -2, -3. For weak fields the ground-state configuration arises from the field-free 1s22s22p02p-1, Sz = -1 configuration. With increasing field strength the ground state involves the four Sz = -2 configurations 1s22s2p02p-12p+1, 1s22s2p02p-13d-2, 1s22p02p-13d-24f-3, and 1s22p-13d-24f-35g-4, followed by the two fully spin-polarized Sz = -3 configurations 1s2p02p-13d-24f-35g-4 and 1s2p-13d-24f-35g-46h-5. The last configuration forms the ground state of the carbon atom in the high-field regime γ > 18.664. The above series of ground-state configurations is extd. from the results of numerical calcns. for more than 20 electronic configurations selected due to some general energetic arguments.**74**Ivanov, M. V.; Schmelcher, P. The boron atom and boron positive ion in strong magnetic fields.*J. Phys. B: At., Mol. Opt. Phys.*2001,*34*, 2031– 2044, DOI: 10.1088/0953-4075/34/10/316Google Scholar74https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXktlSltrs%253D&md5=db05a6fe01708387c70da334cc04331aThe boron atom and boron positive ion in strong magnetic fieldsIvanov, M. V.; Schmelcher, P.Journal of Physics B: Atomic, Molecular and Optical Physics (2001), 34 (10), 2031-2044CODEN: JPAPEH; ISSN:0953-4075. (Institute of Physics Publishing)The ground and a few excited states of the B atom in external uniform magnetic fields are calcd. by a 2-dimensional mesh Hartree-Fock method for field strengths ranging from zero up to 2.35 × 109 T. With increasing field strength the ground state of the B atom undergoes five crossovers involving six different electronic configurations which belong to three groups with different spin projections Sz = -1/2, -3/2, -5/2. For weak fields the ground state configuration arises from the field-free 1s22s22p-1, Sz = -1/2 configuration. With increasing field strength the ground state involves the four Sz = -3/2 configurations 1s22s2p02p-1, 1s22s2p-13d-2, 1s22p02p-13d-2 and 1s22p-13d-24f-3, followed by the fully spin-polarized Sz = -5/2 configuration 1s2p-13d-24f-35g-4. The latter configuration forms the ground state of the B atom in the high-field regime γ > 8.0251. Analogous calcns. for the B+ give a sequence of the four following ground state configurations: 1s22s2 (Sz = 0), 1s22s2p-1 (Sz = -1), 1s22p-13d-2 (Sz = -1) and 1s2p-13d-24f-3 (Sz = -2). The above series of ground state configurations are extd. from the results of numerical calcns. for a no. of electronic configurations selected according to general energetical arguments.**75**Al-Hujaj, O.-A.; Schmelcher, P. Lithium in strong magnetic fields.*Phys. Rev. A: At., Mol., Opt. Phys.*2004,*70*, 033411, DOI: 10.1103/PhysRevA.70.033411Google Scholar75https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXovFGisL4%253D&md5=f26437bde2df5dd07fcbeff0d479505fLithium in strong magnetic fieldsAl-Hujaj, Omar-Alexander; Schmelcher, PeterPhysical Review A: Atomic, Molecular, and Optical Physics (2004), 70 (3), 033411/1-033411/12CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The electronic structure of the lithium atom in a strong magnetic field 0 ≤ γ ≤ 10 is investigated. Our computational approach is a full CI method based on a set of anisotropic Gaussian orbitals that is nonlinearly optimized for each field strength. Accurate results for the total energies and one-electron ionization energies for the ground and several excited states for each of the symmetries 20+, 2(-1)+, 4(-1)+, 4(-1)-, 2(-2)+, 4(-2)+, and 4(-3)+ are presented. The behavior of these energies as a function of the field strength is discussed and classified. Transition wavelengths for linear and circular polarized transitions are presented as well.**76**Berdyugina, S. V.; Berdyugin, A. V.; Piirola, V. Molecular Magnetic Dichroism in Spectra of White Dwarfs.*Phys. Rev. Lett.*2007,*99*, 091101, DOI: 10.1103/PhysRevLett.99.091101Google Scholar76https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXpvVGrsbg%253D&md5=05dbd5cbb45d80548a5b35ff8cf16bf7Molecular Magnetic Dichroism in Spectra of White DwarfsBerdyugina, S. V.; Berdyugin, A. V.; Piirola, V.Physical Review Letters (2007), 99 (9), 091101/1-091101/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)The authors present novel calcns. of the magnetic dichroism appearing in mol. bands in the presence of a strong magnetic field, which perturbs the internal structure of the mol. and results in net polarization due to the Paschen-Back effect. Based on that, the authors analyze new spectropolarimetric observations of the cool magnetic He-rich white dwarf G99-37, which shows strongly polarized mol. bands in its spectrum. In addn. to previously known mol. bands of the C2 Swan and CH A-X systems, the authors find a firm evidence for the violet CH B-X bands at 390 nm and C2 Deslandres-d'Azambuja bands at 360 nm. Combining the polarimetric observations with model calcns., the authors deduce a dipole magnetic field of 7.5 ± 0.5 MG with the pos. pole pointing towards the Earth. The developed technique is an excellent tool for studying magnetic fields on cool magnetic stars.**77**Vornanen, T.; Berdyugina, S. V.; Berdyugin, A. V.; Piirola, V. GJ 841B - The Second DQ White Dwarf with Polarized CH Molecular Bands.*Astrophys. J., Lett.*2010,*720*, L52, DOI: 10.1088/2041-8205/720/1/L52Google ScholarThere is no corresponding record for this reference.**78**Detmer, T.; Schmelcher, P.; Diakonos, F. K.; Cederbaum, L. S. Hydrogen molecule in magnetic fields: The ground states of the Σ manifold of the parallel configuration.*Phys. Rev. A: At., Mol., Opt. Phys.*1997,*56*, 1825– 1838, DOI: 10.1103/PhysRevA.56.1825Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlvVems7o%253D&md5=d12ff91c3ddbefc85db16fb64a48861dHydrogen molecule in magnetic fields: the ground states of the Σ manifold of the parallel configurationDetmer, T.; Schmelcher, P.; Diakonos, F. K.; Cederbaum, L. S.Physical Review A: Atomic, Molecular, and Optical Physics (1997), 56 (3), 1825-1838CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The electronic structure of the hydrogen mol. is investigated for the parallel configuration. The ground states of the Σ manifold are studied for ungerade and gerade parity as well as singlet and triplet states covering a broad regime of field strengths from B = 0 up to B = 100 a.u. A variety of interesting phenomena can be obsd. For the 1Σg state we found a monotonous decrease of the equil. distance and a simultaneous increase of the dissocn. energy with growing magnetic-field strength. The 3Σg state is shown to develop an addnl. min. which has no counterpart in field-free space. The 1Σu state shows a monotonous increase in the dissocn. energy with a first increasing and then decreasing internuclear distance of the min. For this state the dissocn. channel is H2 → H- + H+ for magnetic field strengths B .ltorsim. 20 a.u. due to the existence of strongly bound H- states in strong magnetic fields. The repulsive 3Σu state possesses a very shallow van der Waals min. for magnetic-field strengths smaller than 1.0 a.u. within the numerical accuracy of our calcns. The 1Σg and 3Σu states cross as a function of B and the 3Σu state, which is an unbound state, becomes the ground state of the hydrogen mol. in magnetic fields B .ltorsim. 0.2 a.u. This is of particular interest for the existence of mol. hydrogen in the vicinity of white dwarfs. In superstrong fields the ground state is again a strongly bound state, the 3Πu state.**79**Detmer, T.; Schmelcher, P.; Cederbaum, L. S. Hydrogen molecule in a magnetic field: The lowest states of the Π manifold and the global ground state of the parallel configuration.*Phys. Rev. A: At., Mol., Opt. Phys.*1998,*57*, 1767– 1777, DOI: 10.1103/PhysRevA.57.1767Google Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXhtFOqtrc%253D&md5=b6c8f609ab04b2b1c734796488639525Hydrogen molecule in a magnetic field: The lowest states of the Π manifold and the global ground state of the parallel configurationDetmer, T.; Schmelcher, P.; Cederbaum, L. S.Physical Review A: Atomic, Molecular, and Optical Physics (1998), 57 (3), 1767-1777CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The electronic structure of the hydrogen mol. in a magnetic field is investigated for parallel internuclear and magnetic field axes. The lowest states of the Π manifold are studied for spin singlet and triplet (Ms=-1) as well as gerade and ungerade parity for a broad range of field strengths 0≤B≤100 a.u. For both states with gerade parity we observe a monotonic decrease in the dissocn. energy with increasing field strength up to B=0.1 a.u. and metastable states with respect to the dissocn. into two H atoms occur for a certain range of field strengths. For both states with ungerade parity we observe a strong increase in the dissocn. energy with increasing field strength above some crit. field strength Bc. As a major result we det. the transition field strengths for the crossings among the lowest 1Σg, 3Σu, and 3Πu states. The global ground state for B.ltorsim.0.18 a.u. is the strongly bound 1Σg state. The crossings of the 1Σg with the 3Σu and 3Πu state occur at B≈0.18 and B≈0.39 a.u., resp. The transition between the 3Σu and the 3Πu state occurs at B≈12.3 a.u. Therefore, the global ground state of the hydrogen mol. for the parallel configuration is the unbound 3Σu state for 0.18.ltorsim.B.ltorsim.12.3 a.u. The ground state for B⪆12.3 a.u. is the strongly bound 3Πu state. This result is of great relevance to the chem. in the atmospheres of magnetic white dwarfs and neutron stars.**80**Schmelcher, P. Exploring the topology of potential energy surfaces of the H_{2}^{+}ion in the presence of a strong magnetic field.*Int. J. Quantum Chem.*1997,*64*, 553– 560, DOI: 10.1002/(SICI)1097-461X(1997)64:5<553::AID-QUA6>3.0.CO;2-VGoogle Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlsVyjs7k%253D&md5=8772bf0708335e960c94f6f2dc48b6a3Exploring the topology of potential energy surfaces of the H2+ ion in the presence of a strong magnetic fieldSchmelcher, P.International Journal of Quantum Chemistry (1997), 64 (5), 553-560CODEN: IJQCB2; ISSN:0020-7608. (Wiley)We discuss the symmetries, the behavior of the diabatic energy curves, as well as the static aspects of vibronic interaction for diat. mols. in the presence of a strong magnetic field. Our central subject of investigation is the topol. of the adiabatic electronic potential energy surfaces of diat. mols. which are discussed using some selected examples of the surfaces of the H21+ ion. Global equil. configurations corresponding to stable mol. states are found both for the parallel as well as for the perpendicular configurations. For a higher degree of excitation, we observe that the global min. can belong to the lowest possible symmetry of the ion in the presence of a magnetic field. As an example, we discuss the topol. of the 3u potential energy surface.**81**Augustovičová, L. D.; Špirko, V. Zeeman molecular probe for tests of fundamental physical constants.*Mon. Not. R. Astron. Soc.*2020,*494*, 1675– 1680, DOI: 10.1093/mnras/staa792Google Scholar81https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXis1yktr%252FN&md5=b6c781e4cea284d86ddd15e4328931f4Zeeman molecular probe for tests of fundamental physical constantsAugustovicova, Lucie D.; Spirko, VladimirMonthly Notices of the Royal Astronomical Society (2020), 494 (2), 1675-1680CODEN: MNRAA4; ISSN:1365-2966. (Oxford University Press)The impact of the Zeeman effect on the λ-doublet spectra of diat. radicals is analyzed from the point of view of a possible cosmol. variation of the proton-to-electron mass ratio, ν. The actual model calcns. performed for the 2π3/2 and 2π1/2 states of 16OH reveal that the λ-doublet energy levels of diat. radicals can be tuned to degeneracy by means of the Zeeman effect using realistic magnetic fields. Tuning this degeneracy allows for a dramatic enhancement of the relative mass sensitivity coeffs. of the corresponding transitions and for a substantial redn. of their Doppler broadening. Moreover, unlike their field-free counterparts assocd. with the degeneracies arising due to the A ~ 4B situations (A and B being the spin-orbit and rotation const., resp.), the elec. dipole allowed e↔f Zeeman-tuned transitions exhibit favorable intensities, thus evidencing their promising potential.**82**Labzowsky, L. N.; Lozovik, Y. E. Conjugated molecules in strong magnetic fields.*Int. J. Quantum Chem.*1973,*7*, 985– 989, DOI: 10.1002/qua.560070513Google ScholarThere is no corresponding record for this reference.**83**Luzanov, A. V. Magnetism and the biradicaloid character of π-aromatic and antiaromatic systems in a strong magnetic field.*J. Struct. Chem.*2013,*54*, 277– 282, DOI: 10.1134/S0022476613020017Google Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXotlertbg%253D&md5=0f0ba2223a227256210e35affe15976dMagnetism and the biradicaloid character of π-aromatic and antiaromatic systems in a strong magnetic fieldLuzanov, A. V.Journal of Structural Chemistry (2013), 54 (2), 277-282CODEN: JSTCAM; ISSN:0022-4766. (Springer)The previously developed scheme of the full CI for magnetic perturbations of π systems is transformed into a scheme for calcns. in the finite field. It helps create magnetic portraits of mols., reflecting the essentially nonlinear behavior of conjugated systems in a strong field. In particular, possible latent paramagnetism of arom. systems and correspondingly latent diamagnetism of antiarom. ones is easily detected. The degree of the π electron shell openness as well as the singlet-triplet splitting in the field are evaluated. From the data obtained thus in the strong magnetic field an arom. mol. becomes as a rule biradicaloid and nonarom. Accordingly, an antiarom. system dramatically reduces its initial biradicaloid character and thus loses its antiaromaticity.**84**Bates, J. E.; Furche, F. Harnessing the meta-generalized gradient approximation for time-dependent density functional theory.*J. Chem. Phys.*2012,*137*, 164105, DOI: 10.1063/1.4759080Google Scholar84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhsFGksLrF&md5=d2e6a673d974d619dfe23292efc905b1Harnessing the meta-generalized gradient approximation for time-dependent density functional theoryBates, Jefferson E.; Furche, FilippJournal of Chemical Physics (2012), 137 (16), 164105/1-164105/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functionals within the meta-generalized gradient approxn. (MGGA) are widely used for ground-state electronic structure calcns. However, the gauge variance of the kinetic energy d. τ confounds applications of MGGAs to time-dependent systems, excited states, magnetic properties, and states with strong spin-orbit coupling. Becke and Tao used the paramagnetic c.d. to construct a gauge invariant generalized kinetic energy d. ̂τ. We show that τW ≤ ̂τ, where τW is the von Weizsaecker kinetic energy d. of a one-electron system. Thus, replacing τ by ̂τ leads to current-dependent MGGAs (cMGGAs) that are not only gauge invariant but also restore the accuracy of MGGAs in iso-orbital regions for time-dependent and current-carrying states. The current dependence of cMGGAs produces a vector exchange-correlation (XC) potential in the time-dependent adiabatic Kohn-Sham (KS) equations. While MGGA response properties of current-free ground states become manifestly gauge-variant to second order, linear response properties are affected by a new XC kernel appearing in the cMGGA magnetic orbital rotation Hessian. This kernel reflects the first-order coupling of KS orbitals due to changes in the paramagnetic c.d. and has apparently been ignored in previous MGGA response implementations. Inclusion of the current dependence increases total computation times by less than 50%. Benchmark applications to 109 adiabatic excitation energies using the Tao-Perdew-Staroverov-Scuseria (TPSS) MGGA and its hybrid version TPSSh show that cMGGA excitation energies are slightly lower than the MGGA ones on av., but exhibit fewer outliers. Similarly, the optical rotations of 13 small org. mols. show a small but systematic improvement upon inclusion of the magnetic XC kernel. We conclude that cMGGAs should replace MGGAs in all applications involving time-dependent or current-carrying states. (c) 2012 American Institute of Physics.**85**Schlegel, H. B. Optimization of Equilibrium Geometries and Transition Structures.*Advances in Chemical Physics*; John Wiley & Sons, 1987; pp 249– 286, DOI: 10.1002/9780470142936.ch4 .Google ScholarThere is no corresponding record for this reference.**86**Hratchian, H. P.; Schlegel, H. B.*Theory and Applications of Computational Chemistry*; Elsevier, 2005; pp 195– 249.Google ScholarThere is no corresponding record for this reference.**87**Schlegel, H. B. Geometry optimization.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2011,*1*, 790– 809, DOI: 10.1002/wcms.34Google Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtFWrtrbI&md5=62775a44786f69fbe407f459f9652995Geometry optimizationSchlegel, H. BernhardWiley Interdisciplinary Reviews: Computational Molecular Science (2011), 1 (5), 790-809CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. Geometry optimization is an important part of most quantum chem. calcns. This article surveys methods for optimizing equil. geometries, locating transition structures, and following reaction paths. The emphasis is on optimizations using quasi-Newton methods that rely on energy gradients, and the discussion includes Hessian updating, line searches, trust radius, and rational function optimization techniques. Single-ended and double-ended methods are discussed for transition state searches. Single-ended techniques include quasi-Newton, reduced gradient following and eigenvector following methods. Double-ended methods include nudged elastic band, string, and growing string methods. The discussions conclude with methods for validating transition states and following steepest descent reaction paths.**88**Birkholz, A. B.; Schlegel, H. B. Exploration of some refinements to geometry optimization methods.*Theor. Chem. Acc.*2016,*135*, 84, DOI: 10.1007/s00214-016-1847-3Google ScholarThere is no corresponding record for this reference.**89**Baker, J. Techniques for geometry optimization: A comparison of Cartesian and natural internal coordinates.*J. Comput. Chem.*1993,*14*, 1085– 1100, DOI: 10.1002/jcc.540140910Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtlCmsL0%253D&md5=7eefaf59cb2f3d8be6d14bcd924cbea2Techniques for geometry optimization: a comparison of Cartesian and natural internal coordinatesBaker, JonJournal of Computational Chemistry (1993), 14 (9), 1085-100CODEN: JCCHDD; ISSN:0192-8651.A comparison was made between geometry optimization in Cartesian coordinates (using an appropriate initial Hessian) and in natural internal coordinates. Results on 33 different mols., covering a wide range of symmetries and structural types, demonstrated that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internal coordinates to converge to global min.; whereas, Cartesian optimizations converge to the local min. closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internal coordinated are to be preferred over Z-matrix coordinates. General optimization strategies, using internal coordinates and/or Cartesians, are discussed for both unconstrained and constrained optimization.**90**Baker, J.; Chan, F. The location of transition states: A comparison of Cartesian, Z-matrix, and natural internal coordinates.*J. Comput. Chem.*1996,*17*, 888– 904, DOI: 10.1002/(SICI)1096-987X(199605)17:7<888::AID-JCC12>3.0.CO;2-7Google Scholar90https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XitlyqsLo%253D&md5=b9775b2cace6e28c8e91b2c72fe68b1aThe location of transition states: a comparison of Cartesian, Z-matrix, and natural internal coordinatesBaker, Jon; Chan, ForaJournal of Computational Chemistry (1996), 17 (7), 888-904CODEN: JCCHDD; ISSN:0192-8651. (Wiley)A comparison is made between geometry optimization in Cartesian coordinates, in Z-matrix coordinates, and in natural internal coordinates for the location of transition states. In contrast to the situation with min., where all three coordinate systems are of comparable efficiency if a reliable est. of the Hessian matrix is available at the starting geometry, results for 25 different transition states covering a wide range of structural types demonstrate that in practice Z-matrix coordinates are generally superior. For Cartesian coordinates, the commonly used Hessian update schemes are unable to guarantee preservation of the necessary transition state eigenvalue structure, while current algorithms for generating natural internal coordinates may have difficulty handling the distorted geometries assocd. with transition states. The widely used Eigenvector Following (EF) algorithm is shown to be extremely efficient for optimizing transition states.**91**Broyden, C. G. The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations.*IMA J. Appl. Math*1970,*6*, 76– 90, DOI: 10.1093/imamat/6.1.76Google ScholarThere is no corresponding record for this reference.**92**Fletcher, R. A new approach to variable metric algorithms.*Comput. J.*1970,*13*, 317– 322, DOI: 10.1093/comjnl/13.3.317Google ScholarThere is no corresponding record for this reference.**93**Goldfarb, D. A family of variable-metric methods derived by variational means.*Math. Comput.*1970,*24*, 23– 23, DOI: 10.1090/S0025-5718-1970-0258249-6Google ScholarThere is no corresponding record for this reference.**94**Shanno, D. F. Conditioning of quasi-Newton methods for function minimization.*Math. Comput.*1970,*24*, 647– 647, DOI: 10.1090/S0025-5718-1970-0274029-XGoogle ScholarThere is no corresponding record for this reference.**95**Baker, J. Geometry optimization in Cartesian coordinates: Constrained optimization.*J. Comput. Chem.*1992,*13*, 240– 253, DOI: 10.1002/jcc.540130215Google ScholarThere is no corresponding record for this reference.**96**Baker, J.; Bergeron, D. Constrained optimization in Cartesian coordinates.*J. Comput. Chem.*1993,*14*, 1339– 1346, DOI: 10.1002/jcc.540141111Google Scholar96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXhsFCguw%253D%253D&md5=becce2788711e0ef393d3d0c200b555dConstrained optimization in Cartesian coordinatesBaker, Jon; Bergeron, DoreenJournal of Computational Chemistry (1993), 14 (11), 1339-46CODEN: JCCHDD; ISSN:0192-8651.Modifications are made to a previously published algorithm for constrained optimization in Cartesian coordinates (J. Comp. Chem. 13, 240, 1992) to incorporate both fixed and dummy atoms. Std. distance and angle constraints can now be specified with respect to dummy atoms, greatly extending the range of constraints that can be handled. Fixed atoms can be eliminated from the optimization space and so there is no need to calc. their gradients resulting in potentially significant savings of CPU time in ab initio computations. Several examples illustrate the range and versatility of the modified algorithm.**97**Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.*J. Chem. Phys.*1989,*90*, 1007, DOI: 10.1063/1.456153Google Scholar97https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXksVGmtrk%253D&md5=c6cd67a3748dc61692a9cb622d2694a0Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogenDunning, Thom H., Jr.Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.**98**Irons, T. J. P.; Spence, L.; David, G.; Speake, B. T.; Helgaker, T.; Teale, A. M. Analyzing Magnetically Induced Currents in Molecular Systems Using Current-Density-Functional Theory.*J. Phys. Chem. A*2020,*124*, 1321– 1333, DOI: 10.1021/acs.jpca.9b10833Google Scholar98https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhs12ltLc%253D&md5=1b5b59e765d3dc659bc54ad5e300fbc6Analyzing Magnetically Induced Currents in Molecular Systems Using Current-Density-Functional TheoryIrons, Tom J. P.; Spence, Lucy; David, Gregoire; Speake, Benjamin T.; Helgaker, Trygve; Teale, Andrew M.Journal of Physical Chemistry A (2020), 124 (7), 1321-1333CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)A suite of tools for the anal. of magnetically induced currents is introduced. These are applicable to both the weak-field regime, well described by linear response perturbation theory, and to the strong-field regime, which is inaccessible to such methods. A disk-based quadrature scheme is proposed for the anal. of magnetically induced current susceptibilities, providing quadratures that are consistently defined between different mol. systems and applicable to both planar 2D and general 3D mol. systems in a black-box manner. The applicability of the approach is demonstrated for a range of planar ring systems, the ground and excited states of the benzene mol., and the ring, bowl, and cage isomers of the C20 mol. in the presence of a weak magnetic field. In the presence of a strong magnetic field, the para- to diamagnetic transition of the BH mol. is studied, demonstrating that magnetically induced currents present a visual interpretation of this phenomenon, providing insight beyond that accessible using linear response methods.**99**Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations.*J. Chem. Phys.*2002,*116*, 3175– 3183, DOI: 10.1063/1.1445115Google Scholar99https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhtlSiu7k%253D&md5=0130fa656254a693e80d4be6b0f442b8Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculationsWeigend, Florian; Kohn, Andreas; Hattig, ChristofJournal of Chemical Physics (2002), 116 (8), 3175-3183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The convergence of the second-order Moller-Plesset perturbation theory (MP2) correlation energy with the cardinal no. X is investigated for the correlation consistent basis-set series cc-pVXZ and cc-pV(X+d)Z. For the aug-cc-pVXZ and aug-cc-pV(X+d)Z series the convergence of the MP2 correlation contribution to the dipole moment is studied. It is found that, when d-shell electrons cannot be frozen, the cc-pVXZ and aug-cc-pVXZ basis sets converge much slower for third-row elements then they do for first- and second-row elements. Based on the results of these studies criteria are deduced for the accuracy of auxiliary basis sets used in the resoln. of the identity (RI) approxn. for electron repulsion integrals. Optimized auxiliary basis sets for RI-MP2 calcns. fulfilling these criteria are reported for the sets cc-pVXZ, cc-pV(X+d)Z, aug-cc-pVXZ, and aug-cc-pV(X+d)Z with X=D, T, and Q. For all basis sets the RI error in the MP2 correlation energy is more than two orders of magnitude smaller than the usual basis-set error. For the auxiliary aug-cc-pVXZ and aug-cc-pV(X+d)Z sets the RI error in the MP2 correlation contribution to the dipole moment is one order of magnitude smaller than the usual basis set error. Therefore extrapolations towards the basis-set limit are possible within the RI approxn. for both energies and properties. The redn. in CPU time obtained with the RI approxn. increases rapidly with basis set size. For the cc-pVQZ basis an acceleration by a factor of up to 170 is obsd.**100**Jost, R. Magnetic field control of molecular dissociation energies.*Int. J. Quantum Chem.*1997,*64*, 571– 580, DOI: 10.1002/(SICI)1097-461X(1997)64:5<571::AID-QUA8>3.0.CO;2-TGoogle Scholar100https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlsVyis74%253D&md5=8459496e23a9a92a0ad9579374fadbd9Magnetic field control of molecular dissociation energiesJost, RemyInternational Journal of Quantum Chemistry (1997), 64 (5), 571-580CODEN: IJQCB2; ISSN:0020-7608. (Wiley)We show that it is possible to control the dissocn. energies of mols. with an external magnetic field. We focus our interest on the lowest dissocn. channel for which the two at. and/or mol. products are formed in their ground state. The crucial requirement is the paramagnetic character of at least one of the two dissocn. products. Then, an external magnetic field lowers the energy of the paramagnetic species in its lowest Zeeman component and, possibly, the corresponding energy of dissocn. of the parent mol. This it true for diat. mols. when at least one of the atoms has an odd no. of electrons. This is also true for oxygen and phosphorus atoms which have a 3P2 ground state. The Zeeman energy shift of paramagnetic species is always of the order of 1 cm-1 per T. The main theor. difficulty is to det. the correlation diagram existing between the bound states of the parent mol. and the states of the products, or equivalently, how the energy evolves as a function of the internuclear distance corresponding to the dissocn. coordinate. Little is known about this evolution, except for diat. mols., because the large internuclear distances are difficult to observe exptl. The main part of the information come from ab initio calcns. For diat. mols., the dissocn. coordinate is also the unique internuclear distance while for polyat. mols., the potential energy surface has 3N - 6 coordinates and multidimensional effects should be considered. In any case, the singlet-triplet-quintet, etc... (or doublet-quartet, etc...) interactions should play an important role in the correlation diagram because crossings are expected between singlet and triplet potential energy curves (from short to long internuclear distances) and these interactions transform the crossings into anticrossings. The specific examples of alkali diat. mols. (Li2, Na2, etc...), of NO2 and of (O2)2 are analyzed in details.**101**Runge, K.; Sabin, J. R. Electronic properties of H_{2}^{+}, H_{2}, and LiH in high magnetic fields.*Int. J. Quantum Chem.*1997,*64*, 561– 570, DOI: 10.1002/(SICI)1097-461X(1997)64:5<561::AID-QUA7>3.0.CO;2-UGoogle Scholar101https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXlsVyjsbw%253D&md5=a328cd139be07817ff9c348526ac0941Electronic properties of H2+, H2, and LiH in high magnetic fieldsRunge, Keith; Sabin, John R.International Journal of Quantum Chemistry (1997), 64 (5), 561-570CODEN: IJQCB2; ISSN:0020-7608. (Wiley)Advances in magnet construction technol. have made magnets available with continuous fields of nearly 50 T and with bores of sufficient diam. for expts. In addn. to these magnets, already in use at the National High Magnetic Field Lab. (NHMFL), semicontinuous pulsed sources of 100 T are anticipated in the near future. At its Los Alamos campus, the NHMFL has detonated pulsed magnets of over 1000 T. It thus becomes possible to investigate the behavior of mols. in strong fields with an eye to field-induced changes in such quantities as geometrical and electronic properties, spectroscopic properties, and reactivities. Theory is a useful probe for these quantities and serves to screen among possible candidates for expts. In this contribution, we report preliminary results on calcns. of electronic properties of H21+, H2, and LiH, the simplest of mols. Initial indications are that for increasing applied field strength, mol. bond lengths decrease and binding energies increase, with a concomitant increase in vibrational frequencies. Field-induced changes in these quantities, as well as in ground-state mol. potential energy surfaces are discussed, and suggestions are made for further investigations, both theor. and exptl.**102**Ceulemans, A. J.*Group Theory Applied to Chemistry*; Springer: Dordrecht, The Netherlands, 2013.Google ScholarThere is no corresponding record for this reference.**103**Lange, K. K.; Tellgren, E. I.; Hoffmann, M. R.; Helgaker, T. A Paramagnetic Bonding Mechanism for Diatomics in Strong Magnetic Fields.*Science*2012,*337*, 327– 331, DOI: 10.1126/science.1219703Google Scholar103https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtVOisrrP&md5=9955432d321e69570d849c93efd064eeA Paramagnetic Bonding Mechanism for Diatomics in Strong Magnetic FieldsLange, Kai K.; Tellgren, E. I.; Hoffmann, M. R.; Helgaker, T.Science (Washington, DC, United States) (2012), 337 (6092), 327-331CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)Elementary chem. distinguishes two kinds of strong bonds between atoms in mols.: the covalent bond, where bonding arises from valence electron pairs shared between neighboring atoms, and the ionic bond, where transfer of electrons from one atom to another leads to Coulombic attraction between the resulting ions. We present a third, distinct bonding mechanism: perpendicular paramagnetic bonding, generated by the stabilization of antibonding orbitals in their perpendicular orientation relative to an external magnetic field. In strong fields such as those present in the atms. of white dwarfs (on the order of 105 teslas) and other stellar objects, our calcns. suggest that this mechanism underlies the strong bonding of H2 in the 3Σu+(1σg1σu*) triplet state and of He2 in the 1Σg+(1σg21σu*2) singlet state, as well as their preferred perpendicular orientation in the external field.**104**Austad, J.; Borgoo, A.; Tellgren, E. I.; Helgaker, T. Bonding in the helium dimer in strong magnetic fields: the role of spin and angular momentum.*Phys. Chem. Chem. Phys.*2020,*22*, 23502– 23521, DOI: 10.1039/D0CP03259JGoogle Scholar104https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhvFKntLfL&md5=a9ba655c43d1edd73b228c1aae357538Bonding in the helium dimer in strong magnetic fields: the role of spin and angular momentumAustad, Jon; Borgoo, Alex; Tellgren, Erik I.; Helgaker, TrygvePhysical Chemistry Chemical Physics (2020), 22 (41), 23502-23521CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We investigate the helium dimer in strong magnetic fields, focusing on the spectrum of low-lying electronic states and their dissocn. curves, at the full configuration-interaction level of theory. To address the loss of cylindrical symmetry and angular momentum as a good quantum no. for nontrivial angles between the bond axis and magnetic field, we introduce the almost quantized angular momentum (AQAM) and show that it provides useful information about states in arbitrary orientations. In general, strong magnetic fields dramatically rearrange the spectrum, with the orbital Zeeman effect bringing down states of higher angular momentum below the states with pure σ character as the field strength increases. In addn., the spin Zeeman effect pushes triplet states below the lowest singlet; in particular, a field of one at. unit is strong enough to push a quintet state below the triplets. In general, the angle between the bond axis and the magnetic field also continuously modulates the degree of σ, π, and δ character of bonds and the previously identified perpendicular paramagnetic bonding mechanism is found to be common among excited states. Electronic states with preferred skew field orientations are identified and rationalized in terms of permanent and induced electronic currents.**105**Chu, S.-I.; Yoshimine, M.; Liu, B. Ab initio study of the*X*^{2}Π and*A*^{2}Σ^{+}states of OH. I. Potential curves and properties.*J. Chem. Phys.*1974,*61*, 5389– 5395, DOI: 10.1063/1.1681891Google Scholar105https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2MXktFGqt70%253D&md5=a0a1fba770c15b08afda9cdef185eefcAb initio study of the X2Π and A2Σ+ states of the hydroxyl radical. I. Potential curves and propertiesChu, Shih-I; Yoshimine, M.; Liu, B.Journal of Chemical Physics (1974), 61 (12), 5389-95CODEN: JCPSA6; ISSN:0021-9606.The CI wave functions, potential-energy curves, and 1-electron properties are presented. The calcd. equil. internuclear sepn. (Re, in bohrs), dissocn. energy (De, in eV), and dipole moment (μ, in D) in the v = 0 vibrational state, resp., are: OH(X2π), 1.841, 4.43, 1.634; OH(A2Σ+), 1.906, 2.29, 1.875. Calcd. values are also given for OD. The spectroscopic consts. for OH and OD calcd. from the theor. potential curves agree satisfactorily with the available exptl. data. The other mol. properties calcd. include the quadrupole moments and the elec. field gradients at the nuclei.**106**Qin, X.; Zhang, S. D. Low-lying electronic states of the OH radical: Potential energy curves, dipole moment functions, and transition probabilities.*J. Korean Phys. Soc.*2014,*65*, 2017– 2022, DOI: 10.3938/jkps.65.2017Google Scholar106https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhs1eiurk%253D&md5=18765e7c1b96714b563a5d02fe850afeLow-lying electronic states of the OH radical: Potential energy curves, dipole moment functions, and transition probabilitiesQin, X.; Zhang, S. D.Journal of the Korean Physical Society (2014), 65 (12), 2017-2022CODEN: JKPSDV; ISSN:0374-4884. (Korean Physical Society)The six doublet and the two quartet electronic states (2Σ+(2), 2Σ-, 2Π(2), 2Δ, 4Σ-, and 4Π) of the OH radical have been studied using the multi-ref. CI (MRCI) method where the Davidson correction, core-valence interaction and relativistic effect are considered with large basis sets of aug-cc-pv5z, aug-cc-pcv5z, and cc-pv5z-DK, resp. Potential energy curves (PECs) and dipole moment functions are also calcd. for these states for internuclear distances ranging from 0.05 nm to 0.80 nm. All possible vibrational levels and rotational consts. for the bound state X2Π and A2Σ+ of OH are predicted by numerical solving the radial Schrodinger equation through the Level program, and spectroscopic parameters, which are in good agreements with exptl. results, are obtained. Transition dipole moments between the ground state X2Π and other excited states are also computed using MRCI, and the transition probability, lifetime, and Franck-Condon factors for the A2Σ+-X2Π transition are discussed and compared with existing exptl. values.**107**Maeda, K.; Wall, M. L.; Carr, L. D. Hyperfine structure of the hydroxyl free radical (OH) in electric and magnetic fields.*New J. Phys.*2015,*17*, 045014, DOI: 10.1088/1367-2630/17/4/045014Google ScholarThere is no corresponding record for this reference.**108**Bartlett, R. J.; Stanton, J. F. Applications of Post-Hartree–Fock Methods: A Tutorial. In*Reviews in Computational Chemistry*; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH: New York, 1994; Vol. 5, pp 65− 169.Google ScholarThere is no corresponding record for this reference.**109**Gilbert, A. T. B.; Besley, N. A.; Gill, P. M. W. Self-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)†.*J. Phys. Chem. A*2008,*112*, 13164– 13171, DOI: 10.1021/jp801738fGoogle Scholar109https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtValurbL&md5=3baaf7b15c1c6fcd86bc3c071deacfadSelf-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)Gilbert, Andrew T. B.; Besley, Nicholas A.; Gill, Peter M. W.Journal of Physical Chemistry A (2008), 112 (50), 13164-13171CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present a simple algorithm, which we call the max. overlap method (MOM), for finding excited-state solns. to SCF equations. Instead of using the aufbau principle, the algorithm maximizes the overlap between the occupied orbitals on successive SCF iterations. This prevents variational collapse to the ground state and guides the SCF process toward the nearest, rather than the lowest energy, soln. The resulting excited-state solns. can be treated in the same way as the ground-state soln. and, in particular, derivs. of excited-state energies can be computed using ground-state code. We assess the performance of our method by applying it to a variety of excited-state problems including the calcn. of excitation energies, charge-transfer states, and excited-state properties.**110**Besley, N. A.; Gilbert, A. T. B.; Gill, P. M. W. Self-consistent-field calculations of core excited states.*J. Chem. Phys.*2009,*130*, 124308, DOI: 10.1063/1.3092928Google Scholar110https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXjvVemu7c%253D&md5=2b14b2b3b6f825581ee600b9872e278cSelf-consistent-field calculations of core excited statesBesley, Nicholas A.; Gilbert, Andrew T. B.; Gill, Peter M. W.Journal of Chemical Physics (2009), 130 (12), 124308/1-124308/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The accuracy of core excitation energies and core electron binding energies computed within a Δself-consistent-field framework is assessed. The variational collapse of the core excited state is prevented by maintaining a singly occupied core orbital using an overlap criterion called the max. overlap method. When applied to a wide range of small org. mols., the resulting core excitation energies are not systematically underestimated as obsd. in time-dependent d. functional theory and agree well with expt. The accuracy of this approach for core excited states is illustrated by the calcn. of the pre-edge features in x-ray absorption spectra of plastocyanin, which shows that accurate results can be achieved with Δself-consistent-field calcns. when used in conjunction with uncontracted basis functions. (c) 2009 American Institute of Physics.**111**Barca, G. M. J.; Gilbert, A. T. B.; Gill, P. M. W. Simple Models for Difficult Electronic Excitations.*J. Chem. Theory Comput.*2018,*14*, 1501– 1509, DOI: 10.1021/acs.jctc.7b00994Google Scholar111https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXivVaht7c%253D&md5=ca2b971e3071f917108202889deb5addSimple Models for Difficult Electronic ExcitationsBarca, Giuseppe M. J.; Gilbert, Andrew T. B.; Gill, Peter M. W.Journal of Chemical Theory and Computation (2018), 14 (3), 1501-1509CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a single-determinant approach to three challenging topics in the chem. of excited states: double excitations, charge-transfer states, and conical intersections. The results are obtained by using the Initial Maximum Overlap Method (IMOM) which is a modified version of the Maximum Overlap Method (MOM). The new algorithm converges better than the original, esp. for these difficult problems. By considering several case studies, we show that a single-determinant framework provides a simple and accurate alternative for modeling excited states in cases where other low-cost methods, such as CIS and TD-DFT, either perform poorly or fail completely.**112**Burton, H. G. A.; Thom, A. J. W. Holomorphic Hartree-Fock Theory: An Inherently Multireference Approach.*J. Chem. Theory Comput.*2016,*12*, 167– 173, DOI: 10.1021/acs.jctc.5b01005Google Scholar112https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvVOrt77P&md5=94c0f990e9c94212c29b68ebb6ec078fHolomorphic Hartree-Fock Theory: An Inherently Multireference ApproachBurton, Hugh G. A.; Thom, Alex J. W.Journal of Chemical Theory and Computation (2016), 12 (1), 167-173CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We investigate the existence of holomorphic Hartree-Fock solns. using a revised SCF algorithm. We use this algorithm to study the Hartree-Fock solns. for H2 and H42+ and report the emergence of holomorphic solns. at points of symmetry breaking. Finally, we find these holomorphic solns. for H4 and use them as a basis for Non-Orthogonal CI at a range of rectangular geometries and show them to produce energies in good agreement with Full CI.**113**Baerends, E.; Branchadell, V.; Sodupe, M. Atomic reference energies for density functional calculations.*Chem. Phys. Lett.*1997,*265*, 481– 489, DOI: 10.1016/S0009-2614(96)01449-2Google Scholar113https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXnvVKnsw%253D%253D&md5=fe2d05e516372a960660c19551bd95f0Atomic reference energies for density functional calculationsBaerends, E. J.; Branchadell, V.; Sodupe, M.Chemical Physics Letters (1997), 265 (3-5), 481-489CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)At. ground states are usually degenerate. It is demonstrated that the d. functionals for the exchange-correlation energy that are commonly used are not invariant over the set of ground state densities. This leads to uncertainties of the order of 3 to 5 kcal/mol in the at. ground state energy of second and third period main group elements and the first transition series. A much larger spread in energies is obtained for transition elements if symmetry and equivalence restrictions for the Kohn-Sham orbitals are abandoned. It is recommended that at. ground states that are actually used to calc. heats of atomization are made explicit, and tables with one choice of at. ground state energies for the first rows of the periodic system are provided for the local d. approxn. and for a few generalized gradient approxns.**114**Caputo, M. C.; Lazzeretti, P. Geometry distortion of the benzene molecule in a strong magnetic field.*Int. J. Quantum Chem.*2011,*111*, 772– 779, DOI: 10.1002/qua.22812Google Scholar114https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhsF2mt77E&md5=e51b639aa968e3bc6260f96ca42f577eGeometry distortion of the benzene molecule in a strong magnetic fieldCaputo, M. C.; Lazzeretti, P.International Journal of Quantum Chemistry (2011), 111 (4), 772-779CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The electrostatic Lorentz force acting on the H and C nuclei of a benzene mol. in the presence of a strong magnetic field with flux d. B has been estd. via Rayleigh-Schrodinger perturbation theory to second order in B. In stationary conditions, a new equil. configuration is reached, at which the total force has been entirely transferred to the nuclei, and the force on the electrons vanishes. The distortion of the mol. geometry is rationalized in terms of third-rank elec. hypershielding at the nuclei, induced by strong magnetic fields applied along three Cartesian axes. The nuclear hypershielding has been evaluated at near Hartree-Fock level of accuracy by its definition within the Rayleigh-Schrodinger perturbation theory, and by a pointwise procedure for the geometrical derivs. of magnetic susceptibilities. The connection between these two quantities is provided by the Hellmann-Feynman theorem. A field along the C6 symmetry axis causes a sym. contraction of the carbon ring and an elongation of the CH bonds. A field along one of the C2 symmetry axes contg. two CH bond acts to shorten them, to widen the ring, and to bend the four remaining CH bonds towards C2. A field along one of the C symmetry axes through the midpoint of two opposite CC bonds causes a spindle effect, by squeezing the mol. toward the center of mass. Constraints for rotational and translational invariance and hypervirial theorems provide a natural criterion for Hartree-Fock quality of computed nuclear elec. hypershielding. However, the mol. distortion is negligible for applied fields usually available in a lab. ⊗ 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011.**115**Kuchitsu, T.; Okuda, J.; Tachikawa, M. Evaluation of molecular integral of Cartesian Gaussian type basis function with complex-valued center coordinates and exponent via the McMurchie-Davidson recursion formula and its application to electron dynamics.*Int. J. Quantum Chem.*2009,*109*, 540– 548, DOI: 10.1002/qua.21813Google ScholarThere is no corresponding record for this reference.**116**Kawashima, Y.; Ishimura, K.; Shiga, M. Ab initio quantum mechanics/molecular mechanics method with periodic boundaries employing Ewald summation technique to electron-charge interaction: Treatment of the surface-dipole term.*J. Chem. Phys.*2019,*150*, 124103, DOI: 10.1063/1.5048451Google Scholar116https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXmtVClt7s%253D&md5=1b04a92a4d34298bb2e87817f18239bfAb initio quantum mechanics/molecular mechanics method with periodic boundaries employing Ewald summation technique to electron-charge interaction: Treatment of the surface-dipole termKawashima, Y.; Ishimura, K.; Shiga, M.Journal of Chemical Physics (2019), 150 (12), 124103/1-124103/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have developed a combined quantum mechanics/mol. mechanics (QM/MM) method with periodic boundary condition (PBC) treatment of explicit electron-charge interactions in a theor. rigorous manner, for an accurate description of electronic structures for mols. in the condensed phase. The Ewald summation technique is employed for the calcn. of the one-electron Hamiltonian in an ab initio framework. We decomp. the Coulomb interactions into two components: those within the same cell and those between different cells. The former is calcd. in the same way as the conventional QM/MM calcn. for isolated systems; this article focuses on our novel method for calcg. the latter type of Coulomb interactions. The detailed formulation of the Hamiltonian of this new QM/MM-PBC method, as well as the necessary one-electron integrals and their gradients, is given. The novel method is assessed by applying it to the dil. water system and a system with a coumarin mol. in water solvent; it successfully reproduces the electronic energies, frontier orbital energies, and Mulliken population charge of the real-space limit calcd. by QM/MM using large isolated systems. We investigated the contribution from each term of the Hamiltonian and found that the surface-dipole term in the Ewald summation technique is indispensable for QM/MM-PBC calcns. The newly developed QM/MM-PBC method is promising for tackling chem. reactions and excited states of mols. in the condensed phase. (c) 2019 American Institute of Physics.**117**Flocke, N. On the use of shifted Jacobi polynomials in accurate evaluation of roots and weights of Rys polynomials.*J. Chem. Phys.*2009,*131*, 064107, DOI: 10.1063/1.3204437Google Scholar117https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtVSiu7zI&md5=37a6d3d47835cd831b8e17ad2e42bfc7On the use of shifted Jacobi polynomials in accurate evaluation of roots and weights of Rys polynomialsFlocke, N.Journal of Chemical Physics (2009), 131 (6), 064107/1-064107/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Shifted Jacobi polynomials Gn(p,q,x) can be used in connection with the Gaussian quadrature modified moment technique to greatly enhance the accuracy of evaluation of Rys roots and wts. used in Gaussian integral evaluation in quantum chem. A general four-term inhomogeneous recurrence relation is derived for the shifted Jacobi polynomial modified moments over the Rys wt. function e-Tx/x. It is shown that for q = 1/2 this general four-term inhomogeneous recurrence relation reduces to a three-term p-dependent inhomogeneous recurrence relation. Adjusting p to proper values depending on the Rys exponential parameter T, the method is capable of delivering highly accurate results for large no. of roots and wts. in the most difficult to treat intermediate T range. Examples are shown, and detailed formulas together with practical suggestions for their efficient implementation are also provided. (c) 2009 American Institute of Physics.

## Supporting Information

## Supporting Information

ARTICLE SECTIONSSI The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.0c01297.

Potential energy curves and optimized geometries of OH in 0.1B

_{0}and 0.2B_{0}magnetic fields parallel and perpendicular to the bond, computed with Hartree−Fock; plot of the energies of OH dissociation products with varying field strength computed with Hartree−Fock; optimized structures of benzene with*M*_{s}= 0, −1, −2, and −3 in a 0.1B_{0}magnetic field computed with Hartree−Fock; plots of optimized C−C and C−H bond lengths in benzene with*M*_{s}= 0 and −3 with varying field strength (PDF)

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