**Cite This:**

*J. Chem. Theory Comput.*2024, 20, 20, 8862-8875

# DMRG-Tailored Coupled Cluster Method in the 4c-Relativistic Domain: General Implementation and Application to the NUHFI and NUF_{3} MoleculesClick to copy article linkArticle link copied!

- Jakub VišňákJakub VišňákJ. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech RepublicFaculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague, Czech RepublicMiddle East Technical University, Üniversiteler Mahallesi, Dumlupınar Bulvarı No:1, 06800 Çankaya Ankara, TürkiyeMore by Jakub Višňák
- Jan BrandejsJan BrandejsJ. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech RepublicFaculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague, Czech RepublicFaculty of Science, Humanities, and Education, Technical University of Liberec, Studentská 1402/2, 461 17 Liberec, Czech RepublicMore by Jan Brandejs
- Mihály MátéMihály MátéStrongly Correlated Systems “Lendület” Research Group, Wigner Research Centre for Physics, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, HungaryDepartment of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, GermanyMore by Mihály Máté
- Lucas VisscherLucas VisscherDepartment of Chemistry and Pharmaceutical Sciences, De Boelelaan 1108, Vrije Universiteit Amsterdam, NL-1081 HZ Amsterdam, NetherlandsMore by Lucas Visscher
- Örs Legeza
*****Örs LegezaStrongly Correlated Systems “Lendület” Research Group, Wigner Research Centre for Physics, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, HungaryInstitute for Advanced Study, Technical University of Munich, Lichtenbergstrasse 2a, 85748 Garching, Germany*****E-mail: [email protected]More by Örs Legeza - Jiří Pittner
*****Jiří PittnerJ. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech Republic*****E-mail: [email protected]More by Jiří Pittner

## Abstract

Heavy atom compounds represent a challenge for computational chemistry due to the need for simultaneous treatment of relativistic and correlation effects. Often such systems also exhibit strong correlation, which hampers the application of perturbation theory or single-reference coupled cluster (CC) methods. As a viable alternative, we have proposed externally correcting the CC method using the density matrix renormalization group (DMRG) wave functions, yielding the DMRG-tailored CC method. In a previous paper [*J. Chem. Phys.* **2020**, *152*, 174107], we reported a first implementation of this method in the relativistic context, which was restricted to molecules with real double group symmetry. In this work, we present a fully general implementation of the method, covering complex and quaternion double groups as well. The 4c-TCC method thus becomes applicable to polyatomic molecules, including heavy atoms. For the assessment of the method, we performed calculations of the chiral uranium compound NUHFI, which was previously studied in the context of the enhancement of parity violation effects. In particular, we performed calculations of a cut of the potential energy surface of this molecule along the stretching of the N–U bond, where the system exhibits strong multireference character. Since there are no experimental data for NUHFI, we have performed also an analogous study of the (more symmetric) NUF_{3} molecule, where the vibrational frequency of the N–U bond can be compared with spectroscopic data.

This publication is licensed under

### 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.

### 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.

## I. Introduction

_{2}laser (900–1100 cm

^{–1}), which corresponds to the stretch of the U≡N triple bond in this molecule. On the other hand, stretching a triple bond beyond the immediate vicinity of equilibrium quicky introduces appreciable nondynamic correlation, and the N

_{2}molecule has been often employed as a benchmark to test the performance of multireference correlated methods (20,28−35) and to demonstrate the failure of single-reference ones to properly describe dissociation of this bond. (36) This has motivated us to choose the NUHFI molecule as a relativistic variant of such a benchmark and to study the cut of the potential energy surface of this molecule along the stretch of the N≡U bond (similar studies using pCCD and pCCD-tailored CC have been done by Leszczyk et al. (37)). Since there are no experimental data available for this molecule, we have also performed a study of the NUF

_{3}molecule, which has an analogous N–U bond and where vibrational frequency and anharmonicity have been measured. However, due to its higher symmetry, this molecule does not fully exploit our most general implementation of the 4c-TCCSD method.

## II. Theory and Implementation

*P*,

*Q*,

*R*, and

*S*run over the positive-energy four-component spinors spanning the one-electron basis and ⟨

*PQ*∥

*RS*⟩ = ⟨

*PQ*|

*RS*⟩ – ⟨

*PQ*|

*SR*⟩ stands for the antisymmetrized two-electron integral in Physicist’s notation. In the Kramers-restricted formalism, barred spinors (ϕ

_{p̅}) and unbarred spinors (ϕ

_{p}) form Kramers pairs related to each other by action of the time-reversal operator

*K*:

*M*

_{S}good quantum number is replaced by the

*M*

_{K}quasi quantum number, (39) which is 1/2 for unbarred spinors (A) and −1/2 for spinors with barred indices (B). The capital indices in eq 1 run over both A and B components of Kramers pairs. In contrast to the nonrelativistic case, the Hamiltonian (eq 1) is in general not block-diagonal in

*M*

_{K}, but this number can then serve to partition the operators and Hamiltonian blocks in much the same way as is done with the

*M*

_{S}quantum number. Since each creation or annihilation operator in eq 1 changes

*M*

_{K}by ±1/2, the Hamiltonian can only directly couple states with |Δ

*M*

_{K}| ≤ 2. States with a still higher difference in |

*M*

_{K}| are only coupled indirectly and can possibly be neglected if the Kramers’ pairing is chosen such that it minimizes the couplings between states with different

*M*

_{K}values. Also with the aid of Kramers’ symmetry, the index permutation symmetry of the 2e integrals in eq 1 is still lower than in the nonrelativistic case as this is based on the use of real-valued spin–orbitals, whereas relativistic spinors are in general complex-valued.

*D*

_{2h}

^{*}and subgroups). (42) The binary double groups can be divided into three classes based on the Frobenius–Schur indicator: “real groups” (

*D*

_{2h}

^{*},

*D*

_{2}

^{*}, and

*C*

_{2v}

^{*}); “complex groups” (

*C*

_{2h}

^{*},

*C*

_{2}

^{*}, and

*C*

_{s}

^{*}); and “quaternion groups” (

*C*

_{i}

^{*}and

*C*

_{1}

^{*}). (43) Generalization of nonrelativistic post-HF methods is simplest for the “real groups”, where the integrals in eq 1 are real-valued and the ones with an odd number of barred (B) indices vanish; i.e., one has only to include additional “spin cases” of integrals ⟨AA|BB⟩ and ⟨AB|BA⟩ (in Physicist’s notation) as well as take care of the fact that in general integrals ⟨BA|BA⟩ ≠ ⟨AA|AA⟩. For the complex groups, the integrals are complex-valued, but still only integrals with an even number of barred indices are nonzero. Finally, in the lowest symmetry case of “quaternion groups,” all the integrals have to be included and are complex-valued. (43−46)

*T*

_{cas}containing amplitudes with all active indices is “frozen” at values obtained from CASCI or in our case from DMRG. The external cluster operator

*T*

_{ext}is composed of amplitudes with at least one index outside the DMRG active space.

*T*

_{cas}and

*T*

_{ext}to single and double excitations. The excitation operators

*T*

_{ext}and

*T*

_{cas}commute, which allows the use of the standard CCSD solver, modified to keep the amplitudes from

*T*

_{cas}fixed at values obtained from the DMRG MPS wave function. (53) Since the Hamiltonian contains only one- and two-body terms, the TCCSD energy with

*T*

_{ext}= 0 and

*T*

_{cas}from DMRG reproduces the DMRG energy. In the limit of the DMRG active space including all MOs, TCC thus recovers the FCI energy. In general, a quadratic error bound valid for TNS-TCC methods has been derived. (19)

^{–12}. Rounding has been introduced at various points of the algorithm, and finally, we achieved a stable version which is more efficient than the Lánczos once a good starting vector together with preconditioning is provided. Alternatively, the Lánczos algorithm can be used which requires more iteration steps in the course of the diagonalization but provides very stable convergence even in the case of small numerical noise in the real and imaginary components. As a hybrid solution, we use the Lánczos algorithm during the warmup procedure via the dynamic extended active space procedure (DEAS) (53) and the Davidson method for all further DMRG sweeps once initial starting vectors for the diagonalization step is available by the DMRG wave function transformation protocol. (5) Similar developments have been done on the post-DMRG utility functions, i.e., calculation of single orbital entropy, mutual information, and obtaining the T1 and T2 amplitudes. The full workflow of the general 4c-TCC in the case of quaternion symmetry has been tested and benchmarked against full-CI reference data on small systems. For larger system sizes, however, the methods for optimization of site ordering turned out to be a delicate issue and the procedure from the nonrelativistic DMRG turned out in general not to be transferable. Therefore, particular care has to be taken to the ordering of the sites, which has to keep the A and B spinors from each Kramers pair adjacent. We used thus the order of adjacent pairs of A and B spinors sorted by HF orbital energy, which is probably not optimal, but the optimization was not able to improve it. Besides the complex-valued arithmetics and larger number of integrals, this also contributed to the high computational cost of the 4c-DMRG calculations.

*c*

_{i}

^{a}and

*c*

_{ij}

^{ab}have been extracted from the MPS wave function, the standard CC analysis is performed to convert them to the CC amplitudes

*M*

_{K}. To our advantage, this also matches the single-spinor site representation in the DMRG. In the lowest quaternion-symmetry case, the SD amplitudes are in general all nonzero and complex-valued, while in higher symmetry they become sparse, analogously to the one- and two-electron integrals. Due to the combined effect of doubling the amplitude index range compared to nonrelativistic spin-restricted CCSD and of complex arithmetics required, the low-symmetry relativistic calculations are computationally more demanding by a prefactor of almost 2 orders of magnitude, which restricts the applications to small molecules.

^{–8}. Our implementation is, however, a general one and can easily be extended toward more complex calculations with larger bond dimension and larger active spaces. In fact, from the DMRG viewpoint, recently we have introduced massively parallel hybrid CPU–GPU DMRG building on state of the art software and hardware technologies, (56−59) reaching a quarter petaflops performance and large active spaces even on a single node via applied bond dimension values on the order of tens of thousands. Our implementation has a nearly ideal scaling with the number of computational units and can be extended to several nodes; thus, extension of our hybrid CPU-GPU tensor and matrix libraries to the complex case will boost tremendously the efficiency of 4c-DMRG-TCC in the near future. From the point of view of the TCC, the initial implementation of 4c-CCSD in DIRAC represents a bottleneck, but recently a distributed-GPU implementation of the general-order CC has been reported by DIRAC developers, (60) designed to be compatible with 4c calculations. Therefore, the 4c-TCC method should in the future be reimplemented using this machinery to be able to compute larger molecules. In addition, an efficient construction of amplitudes with higher excitation ranks, like t3, t4, etc., from the DMRG wave function has also been developed and applied already in standard quantum chemistry, which can again be transferred to the complex framework (unpublished yet). Therefore, such developments will be available for the relativistic framework, as well.

*O*(

*D*

^{3}

*N*

^{4}) with just a prefactor on the order of two. This can, however, be accelerated tremendously via AI accelerators as reported in refs (57) and (59). Calculation of the amplitudes from the DMRG wave function using algorithmic solutions outlined in the previous paragraph takes only a fraction of time, usually on the order of only a few minutes even up to CC excitation rank 4. The CC part is performed via DIRAC with taking again a relatively smaller amount of time since the main bottlenecks were the memory requirements. We had to truncate the virtual space for CC to make the calculations tractable, and since CCSD scales as $(\mathcal{O}{N}_{o}^{2}{N}_{v}^{4})$, this truncation shortens the CPU time for CC substantially.

## III. Computational Details

*C*

_{3v}molecular geometry of NUF

_{3}(cf. Figure 1) has been adopted from a CASPT2 computation found in the literature. (61)

*M*= 512 for

*R*

_{NU}= 1.4, 1.7, 2.2, and 4.8 Å. The notation (

*n*,

*m*) denotes

*n*correlated electrons in

*m*Kramers pairs, while other electrons are kept inactive. The amplitude space for both CCSD and TCCSD (47th to 252nd Kramers pair) has been chosen based on the trade-off between computational demands and spectroscopic properties prediction accuracy taking into account energy gaps in the molecular bispinor set.

*z*axis, which is a trivial condition to fulfill. The other relations require the UN mode to be weakly coupled to the other modes. This assumption was verified by checking the normal modes computed at the DFT level of theory. For both molecules considered, the normal mode considered has less than 2% admixture of coordinates other than the

*R*

_{Nz}coordinate. The anharmonicities were computed using the program VIBANAL (65) from a polynomial fit of the potential curves along the UN bond employing the effective mass

^{–12}). However, we checked that no-cutoff does not lead to significantly different results, while it more than doubles the computational time. The same is true for default approximation to the (SS|SS) type of integrals. The default approximation is described by the DIRAC manual as “.LVCORR”, which activates the Dirac–Coulomb Hamiltonian in which (SS|SS) integrals are neglected and replaced by an interatomic SS correction. (40,66) Furthermore, the cutoff applied in the DMRG to the integrals was 10

^{–12}, and the residual error of the Davidson/Lánczos diagonalization procedure was set to 10

^{–8}. The energy changes in the last two DMRG sweeps were on the order of 10

^{–5}.

### III.A. Preparation of the Molecular Spinors

*R*= 1.90 Å for NUHFI) into a lower branch (black line in Figure 3) with a double-well character and an upper branch with a shape typical for the ground state of diatomic molecules (red line). The DHF solution of each branch can be followed by restarting the SCF procedure from a converged solution at a neighboring geometry. Unfortunately, the molecular spinors from either branch lead to unphysical artifacts in the potential energy curves, when a subsequent DMRG calculation with a limited active space is performed. The best solution of this problem would probably be a 4c-CASSCF calculation; unfortunately, we did not find any working implementation of this method

*for quaternion double group symmetry*, neither in DIRAC nor in the BAGEL and ChronusQ codes (in the version available to us). Fortunately, the DFT/PBE N–U potential energy curve (blue line in Figure 3) exhibits no such instability and bifurcation. Even the single-determinant Dirac–Hartree–Fock energy computed from the self-consistently converged PBE Kohn–Sham molecular spinors (green line in Figure 3) is free of bifurcation or double-well problems. However, the HOMO–LUMO gap is significantly narrowing after

*R*> 2.0 Å in these orbitals, which leads to convergence problems of CC methods at larger N–U distances. There are additional reasons for selecting the PBE functional as a source of molecular spinors: We have tried several meta-GGA/hybrid/advanced functionals, and the Hartree–Fock energy potential energy curves computed from their converged spinors exhibited a nonphysical maximum at larger N–U distances (cf. Supporting Information, Figure S1). GGA functionals (including PBE) have been found to be free of this artifact, and from the ones we tested, the PBE spinors provided the lowest Hartree–Fock energy.

## IV. Results and Discussion

### IV.A. NUF_{3} Molecule

_{3}, for which the vibrational frequency of the N–U stretch has been measured. (67) The experimental anharmonicity is not available, so we tried to make a plausible estimate based on the measured anharmonicity 5.3 cm

^{–1}in NU

^{+}(68) with an error bar of ±8%, which corresponds to the spread of the N–U bond vibrational frequencies in the set of related species (UN

^{+}, (68) UN, (69) NUO

^{+}, (69) NUN, (69) NUF

_{3}, (61,69) NUNH (69))

_{3}has always been 35–172 and will be henceforth omitted.

_{3}. Similarly to the NUHFI molecule, uncorrelated results from DHF-SCF and DHF from PBE spinors strongly underestimate the bond length and overestimate the vibrational frequency. The bifurcation of the DHF-SCF solution occurs at

*R*= 1.93 Å, as can be seem from Figure S2 in the Supporting Information.

method | r_{e}, Å | E_{min}, a.u. | ω_{e}, cm^{–1} | ω_{e}x_{e}, cm^{–1} | Δω_{e}, cm^{–1} | Δω_{e}x_{e}, cm^{–1} |
---|---|---|---|---|---|---|

4c-DHF-SCF^{e} | 1.6704 | –28412.17024 | 1282.86 | 5.64 | 344.86 | 0.34 |

4c-PBE | 1.7567 | –28428.26886 | 973.23 | 3.88 | 35.23 | –1.42 |

4c-DHF(PBE) | 1.6553 | –28411.96614 | 1307.93 | 2.98 | 369.93 | –2.32 |

4c-CCSD^{a} | 1.7212 | –28413.59978 | 1106.06 | 3.06 | 168.06 | –2.24 |

4c-CCSD(T)^{a} | 1.7787 | –28413.70853 | 930.2 | 2.74 | –7.80 | –2.56 |

4c-DMRG(14,15) | 1.7565 | –28412.08719 | 805.61 | 3.25 | –132.39 | –2.05 |

4c-DMRG(26,26)^{e} | 1.7777 | –28412.16795 | 821.37 | 6.89 | –116.63 | 1.59 |

4c-DMRG(32,32) | 1.7908 | –28412.23037 | 910.19 | 8.81 | –27.81 | 3.51 |

4c-(14,15)TCCSD^{a} | 1.7661 | –28413.63386 | 923.95 | 2.83 | –14.05 | –2.47 |

4c-(20,32)TCCSD^{a} | 1.7576 | –28413.64760 | 964.93 | 4.47 | 26.93 | –0.83 |

4c-(26,26)TCCSD^{a} | 1.7587 | –28413.64416 | 954.6 | 5.36 | 16.60 | 0.06 |

4c-(32,32)TCCSD^{a} | 1.7493 | –28413.64722 | 1008.88 | 6.89 | 70.88 | 1.59 |

ECP^{b}/CCSD(F12) | 1.6995 | –829.92403 | 1104.63 | 5.34 | 166.63 | 0.04 |

ECP^{b}/CCSD(T)(F12)^{d} | 1.7218 | –829.99853 | 1061.62 | 2.80 | 123.62 | –2.50 |

ECP^{b}/CCSD(T)(F12)^{g} | 1.7200 | –829.99844 | 1058.26 | 4.95 | 120.26 | –0.35 |

ECP^{b}/CCSD(T*)(F12)^{d} | 1.7247 | –830.00965 | 1083.8 | 5.07 | 145.80 | –0.23 |

ECP^{b}/CCSD(T*)(F12)^{g} | 1.7216 | –830.00941 | 1053.39 | 5.59 | 115.39 | 0.29 |

ECP^{b}/DFT:r2SCAN | 1.7093 | –831.52519 | 1000.59 | 3.43 | 62.59 | –1.87 |

ECP^{b}/DFT:mpsts-noa2 | 1.7279 | –831.33322 | 1019.11 | 3.50 | 81.11 | –1.80 |

CASPT2(6,16)^{i} | 1.753 | |||||

exptl.^{j} | 938 | |||||

estimate^{k} | 5.3 ± 0.4 |

^{a}

All 4c-CC and 4c-TCC computations have been done with amplitude range 35–172.

^{b}

Scalar quasi-relativistic, with def2-TZVPP AO basis, computed via Turbomole V7.6. (62,70)

^{d}

The curve is limited by *R*_{max} = 2.00 Å.

^{e}

The curve is limited by *R*_{max} = 1.90 Å.

^{g}

The curve is limited only by *R*_{max} = 2.10 Å, but a higher branch of the bifurcated HF-SCF solution was taken for longer distances of *R* > 1.93 Å.

^{i}

From ref (61).

^{j}

Experimental data from ref (67) indicated in boldface.

^{k}

Estimate based on the experimental anharmonicity value for NU^{+} (68) and a spread of vibrational frequencies in a group of similar molecules.

_{3}has been always 35–172 and will be henceforth be omitted. Concerning the single-reference CC methods, addition of perturbative triples substantially elongates the bond and lowers the vibrational frequency, exemplifying the importance of the dynamic correlation. DMRG consistently yields bond lengths close to the CCSD(T) value, while the vibrational frequencies are too small compared to the experimental value, except in the largest (32,32) space, where ω

_{e}is close to the experiment. However, DMRG in this space yields a too long bond length and an anharmonicity that is too high of an anharmonicity.

^{–1}, with the one for the optimal space (26,26) being 955 cm

^{–1}, which is reasonably close to the experimental value of 938 cm

^{–1}. The anharmonicities are in this case more sensitive to the active space choice than in the NUHFI molecule. Small active spaces yield values close to the single reference methods, while larger spaces yield larger anharmonicity values. For the optimal space (26,26), the result is 5.3 cm

^{–1}, which is in line with the estimate based on the experimental data of similar molecules. (68)

^{–1}, which is close to the estimate. The ECP/DFT results yield similar bond lengths as the CC methods, their vibrational frequencies being substantially over the experimental one and their anharmonicities below the estimate. It thus seems that a combined description of the dynamic and static correlation is needed for this molecule as well in order to describe its spectroscopic properties, particularly the fact that DMRG in modest spaces is not able to yield reliable anharmonicities.

### IV.B. NUHFI Molecule

*r*

_{N–U}= 1.5 Å, while the results for other distances can be found in the Supporting Information. The results for A and B spinors form two almost identical blocks, and the bond dimensions 128 and 256 yield very similar results, indicating sufficient accuracy for this purpose. Based on the entropy profiles, we have chosen three active spaces, small (14,15), medium (24,26), and large (32,32). The orbital entropies yield a quantitative indication of how important individual orbitals are, and within the DMRG method, the more one can afford to include in the active space, the better. However, in the TCC method, a too large active space might not be ideal due to the “freezing” of the static correlation in this space, which does not interact with the dynamical correlation effects outside the active space. As shown previously, (20,26) in the TCC method, a smaller active space may thus yield better results than a bigger one. We have thus also performed an attempt to find the optimum active space for TCCSD, cf. Figure 5, using the DMRG bond dimension

*M*= 1024 and CC orbital space restricted to 47–163. As can be seen, the space (20,28) seems to perform best in terms of yielding the lowest TCC energy at the bond distance 1.7 Å. Of course, at a different bond distance, the outcome might be different, and it is clearly not practical to perform such optimization globally. We have thus confined the detailed calculation of the potential energy curves to the three entropy-selected active spaces and this “TCC-optimal” active space.

*M*= 2048, and in spite of this relatively large bond dimension, the DMRG does not seem close to saturation either. We extrapolated DMRG(32,32,

*M*) energies to infinite bond dimension

*M*using a simple second order polynomial and a three-parameter model:

*E*(

*M*) is the DMRG(32,32,

*M*) energy,

*E*(

*M*→ +∞) its extrapolated value, and

*a*and

*p*are positive fit parameters.

*E*(

*M*) data have been collected from

*M*= 64, 128, 256, 350, 512, 750, 1024, and 2048. (Of course, the bond dimension has a finite maximum value in any finite AO basis, but a large enough to make such an extrapolation sensible.) We have found that the best fit with lowest residual error has been obtained by fixing

*p*= −1/2, which is usually applied to fit energy gaps in gapped systems (to which molecules belong), where the correlations decay exponentially. We note that the extrapolation using a general power

*p*of eq 12 did not lead to significantly different values and was numerically unstable for

*R*> 1.86 Å. The fit via the second order polynomial was very unreliable. A more rigorous fit based on the DMRG truncation error could not be applied here due to the very limited range of the available truncation error data points.

#### IV.B.1. Assessment of TCC in a Small Orbital Space with Respect to Extrapolated DMRG As a Benchmark

*M*→ ∞ limit according to eq 12, which should be equivalent to CASCI(32,32).

*M*= +∞). The vibrational frequency for (24,26)TCCSD agrees well with CCSD(T) while being about 70 cm

^{–1}below the extrapolated DMRG. The (24,26)TCCSD anharmonicity is close to the exptrapolated DMRG, while standard CCSD and CCSD(T) yield lower anharmonicity values. Tailored CC spectroscopic constants are rather sensitive to the active space chosen, which stresses the necessity of appropriate active-space choice for TCCSD. Inside the orbital space (32,32) used as a restriction for both amplitudes and active orbitals, it is not possible to go reasonably much above (24,26), as there will hardly be any amplitudes left. Performing TCC in such a small virtual orbital space, although useful as a sanity check, clearly is not in the spirit of using TCC to capture the dynamic correlation. We attempted a similar test in a larger space corresponding to DMRG(32,98), which was the space used for orbital entropy calculations at low

*M*, but it turned out to be computationally too costly.

method | r_{e}, Å | E_{min}, a.u. | ω_{e}, cm^{–1} | ω_{e}x_{e}, cm^{–1} | Δω_{e}, cm^{–1} | Δω_{e}x_{e}, cm^{–1} |
---|---|---|---|---|---|---|

DMRG(M = 1024, d)^{c} | 1.7591 | –35329.62361 | 1062.42 | 17.19 | –9.35 | 9.95 |

DMRG(M = 1024, s)^{d} | 1.7567 | –35329.62348 | 1053.08 | 19.50 | –18.69 | 12.26 |

DMRG(M = 2048, s)^{d} | 1.7727 | –35329.63243 | 946.67 | 8.92 | –125.10 | 1.68 |

DMRG(M = +∞, s)^{d}^{,}^{e} | 1.7766 | –35329.65525 | 1006.63 | 10.04 | –65.14 | 2.80 |

CCSD | 1.7561 | –35329.61877 | 1084.92 | 5.44 | 13.15 | –1.80 |

CCSD(T) | 1.8110 | –35329.66792 | 929.38 | 3.17 | –142.39 | –4.07 |

(14,15)TCCSD | 1.8088 | –35329.64278 | 812.16 | 2.80 | –259.61 | –4.44 |

(20,26)TCCSD | 1.7680 | –35329.63902 | 998.56 | 22.80 | –73.21 | 15.56 |

(24,26)TCCSD | 1.7852 | –35329.64465 | 935.77 | 10.91 | –136.00 | 3.67 |

^{c}

Dense sampling of distances, with dR = 0.01 Å step. That have been used for all CC methods as well.

^{d}

Spare sampling of distances, with dR = 0.05 Å step.

^{e}

Extrapolation to infinity M done with respect to *E* = *E*(*M* → + ∞) + *a*/√*M* formula and with set starting with *M*_{min} = 64 and all powers of 2 to *M*_{max} = 2048 together with *M* = 350 and *M* = 750

#### IV.B.2. Potential Energy Curves and Spectroscopic Constants

method | r_{e}, Å | E_{min}, a.u. | ω_{e}, cm^{–1} | ω_{e}x_{e}, cm^{–1} | Δω_{e}, cm^{–1} | Δω_{e}x_{e}, cm^{–1} |
---|---|---|---|---|---|---|

4c-DHF-SCF | 1.6614 | –35329.58205 | 1280.23 | 3.59 | 208.46 | –3.65 |

4c-PBE | 1.7408 | –35348.15605 | 1018.67 | 4.43 | –53.10 | –2.81 |

4c-DHF(PBE) | 1.6437 | –35329.36686 | 1338.49 | 3.08 | 266.72 | –4.16 |

4c-B3LYP | 1.7224 | –35346.76292 | 1075.25 | 3.47 | 3.48 | –3.77 |

ECP/B3LYP,TZ^{b} | 1035.0 | 24.00 | ||||

4c-B3LYP,TZ^{c} | 1065.4 | 54.40 | ||||

4c-DMRG(14,15)^{I} | 1.7016 | –35329.46503 | 904.28 | 17.65 | –167.49 | 10.41 |

4c-DMRG(20,28)^{j} | 1.7674 | –35329.58429 | 1001.29 | 4.81 | –70.48 | –2.43 |

4c-DMRG(24,26)^{k} | 1.7547 | –35329.54655 | 847.93 | 3.78 | –223.84 | –3.46 |

4c-DMRG(32,32) | 1.7634 | –35329.62372 | 1060.59 | 7.14 | –11.18 | –0.10 |

4c-CCSD^{g} | 1.7059 | –35331.33308 | 1139.34 | 3.29 | 67.57 | –3.95 |

4c-CCSD(T)^{g} | 1.7604 | –35331.44856 | 973.93 | 2.99 | –97.84 | –4.25 |

4c-(14,15)TCCSD^{g} | 1.7324 | –35331.35627 | 969.2 | 7.63 | –102.57 | 0.39 |

4c-(20,28)TCCSD^{g}^{,}^{h} | 1.7314 | –35331.37442 | 1071.77 | 7.24 | 0.00 | 0.00 |

4c-(24,26)TCCSD^{d}^{,}^{g} | 1.7353 | –35331.06927 | 1036.53 | 6.62 | –35.24 | –0.62 |

4c-(32,32)TCCSD^{g} | 1.7250 | –35331.37310 | 1104.49 | 8.17 | 32.72 | 0.93 |

ECP^{f}/CCSD(F12) | 1.6819 | –642.23964 | 1149.47 | 3.58 | 77.70 | –3.66 |

ECP^{f}/CCSD(T)(F12)^{d} | 1.7061 | –642.30227 | 1095.43 | 2.69 | 23.66 | –4.55 |

ECP^{f}/CCSD(T*)(F12)^{d} | 1.7079 | –642.31204 | 1092.03 | 2.45 | 20.26 | –4.79 |

ECP^{f}/DFT:r2SCAN | 1.7114 | –643.73867 | 1061.35 | 3.38 | –10.42 | –3.86 |

ECP^{f}/DFT:mpsts-noa2 | 1.7056 | –643.51394 | 1077.54 | 3.34 | 5.77 | –3.90 |

^{a}

The parts of potential energy curves delimited by a maximal N–U distance *R*_{max} = 2.100 Å and a minimal *R*_{min} corresponding to the same energy as at *R*_{max} have been used for the fit and calculation of spectroscopic constants, unless indicated otherwise below.

^{b}

Scalar quasi-relativistic computation. From ref (27).

^{c}

Computation From ref (27), harmonic value, four-component Hamiltonian.

^{d}

*R*_{max} = 2.040 Å.

^{f}

Scalar quasi-relativistic, with def2-TZVPP AO basis, computed using Turbomole V7.6. (62,70)

^{g}

4c-CC and almost all 4c-TCC computations have been done with amplitude range 47–252.

^{h}

The bold text corresponds to the optimal active space for the 4c-TCCSD method selected by minimization of 4c-TCCSD energy at *R* = 1.700 Å (see text for more details).

^{I}

Bond dimension *M* = 2048, number of sweeps = 41.

^{j}

*R*_{max} = 1.850 Å.

^{k}

*R*_{max} = 1.880 Å.

^{–1}except for the (20,28) and (32,32) spaces, while the anharmonicities differ over a wide range. Particularly the anharmonicity from the smallest DMRG space (14,15) is an outlier. The reason might be that these spaces are rather small and addition of a few more orbitals has thus a large effect. However, as can be seen on the next lines of the table, the 4c-TCCSD method, although based on DMRG, yields anharmonicity values in a rather narrow range, even for the small space, where DMRG value was extremely large.

^{–1}. Anharmonicities are again less sensitive to the active space, being all around 7 cm

^{–1}. Compared to standard single-reference 4c-CCSD and 4c-CCSD(T), the 4c-TCCSD equilibrium distance and vibrational frequency lies between the 4c-CCSD and 4c-CCSD(T) values. This might be explained by the fact that these properties are determined by the PEC around equilibrium, where the multireference character is still weak and 4c-TCCSD can capture more of the dynamic correlation than 4c-CCSD but less than 4c-CCSD(T). On the other hand, the 4c-TCCSD anharmonicity is approximately twice as large than the 4c-CCSD or 4c-CCSD(T) values, which are similar. Since already the DMRG method alone yielded larger anharmonicities, although widely varying in dependence on the active space, one can deduce that the strong correlation is more important for the anharmonicity, in line with the fact that the multireference character of the system grows with the elongation of the N–U bond. Single-reference methods like CCSD or CCSD(T) thus probably underestimate the anharmonicity. It is possible that DMRG or 4c-TCCSD might on the other hand overestimate it, but for the NUHFI molecule we unfortunately do not have an experimental reference.

## V. Conclusions

_{3}molecules. This bond can be viewed as a relativistic analogue of the triple bond in N

_{2}, which is well-known for its multireference character at stretched bond lengths. The chiral NUHFI molecule is also interesting as a candidate for measurable electroweak parity violating effects, while for NUF

_{3}, experimental vibrational frequency is available.

_{3}vibrational frequency about 17 cm

^{–1}above the experimental value, while the anharmonicity was in line with an estimate based on an experimental value for similar molecules. This agreement was considerably better than the performance of DMRG or single-reference 4c-CCSD alone. For both molecules, the 4c-TCCSD method yielded bond lenghts and vibrational frequencies between the values from single-reference 4c-CCSD and 4c-CCSD(T) methods, indicating that 4c-TCCSD is able to capture more dynamical correlation than 4c-CCSD but less than 4c-CCSD(T). 4c-TCCSD yielded larger anharmonicity values than the single-reference CC methods, which can be attributed to the better description of the potential energy curves at larger distances, where the strong correlation plays a bigger role. The 4c-TCCSD method thus offers benefits compared to single-reference CC methods; however, for future applications the availability of relativistic 4c-CASSCF and/or DMRG-SCF without symmetry restrictions will be important.

## Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.4c00641.

Files for frequency recomputation (ZIP)

Optimized geometries, assessment of XC functionals, ECP computational details, visualization of active molecular orbitals, TCC active space optimization, results in MP2 NO basis, orbital entropy figures (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

The work of the Czech team has been supported by the Czech Science Foundation Grant No. 18-24563S and subsequently by the Advanced Multiscale Materials for Key Enabling Technologies project, supported by the Ministry of Education, Youth, and Sports of the Czech Republic. Project No. CZ.02.01.01/00/22_008/0004558, co-funded by the European Union. Ö.L. has been supported by the Hungarian National Research, Development, and Innovation Office (NKFIH) through Grant No. K134983 and TKP2021-NVA-04; by the Quantum Information National Laboratory of Hungary; by the Hans Fischer Senior Fellowship programme funded by the Technical University of Munich─Institute for Advanced Study; and by the Center for Scalable and Predictive Methods for Excitation and Correlated Phenomena (SPEC), funded as part of the Computational Chemical Sciences Program, FWP 70942, by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences at Pacific Northwest National Laboratory. Mutual visits with the Hungarian group have been partly supported by the Hungarian–Czech Joint Research Project MTA/19/04. J.B. acknowledges the support of a Charles University START programme student grant (No. CZ.02.2.69/0.0/0.0/19_073/0016935). Part of the computational resources were provided by the e-INFRA CZ project (ID:90254), supported by the Ministry of Education, Youth and Sports of the Czech Republic.

## Appendix. Choice of the DFT Functional Employed to Obtain the Molecular Spinor Basis

_{e}could be found (Table 4). We employed three common XC functionals (PBE, BLYP, and B3LYP) and def-TZVPP basis set as implemented in the Turbomole program. (62,70)

system | exper. | ref | B3LYP | BLYP | PBE |
---|---|---|---|---|---|

UN^{+}^{a} | 1083 | (68) | 1094 | 1025 | 1058 |

UN | 1010 | (69) | 1038 | 977 | 1010 |

NUO^{+} | 1017 | (69) | 1182 | 1097 | 1031 |

NUN^{b} | 1051 | (69) | 1116 | 1054 | 1083 |

NUF_{3} | 938 | (61), (69) | 1042 | 934 | 973 |

NUNH | 967 | (69) | 1038 | 979 | 1013 |

MAE^{d} | 74 | 32 | 26 | ||

MQE^{e} | 90 | 43 | 30 | ||

NUHFI | n.a.^{c} | 1073 | 974 | 1011 |

^{a}

For this molecular ion, also experimental anharmonicity ω_{e}*x*_{e} = 5.3 cm^{–1} has been determined. (68)

^{b}

Antisymmetric stretch (σ_{u}), visible in IR spectra.

^{c}

To our best knowledge, NUHFI molecule has not been synthesized yet; there are thus no experimental data.

^{d}

Mean absolute error.

^{e}

Square root of mean of squares of errors.

## References

This article references 73 other publications.

**1**White, S. R. Density matrix formulation for quantum renormalization groups.*Phys. Rev. Lett.*1992,*69*, 2863– 2866, DOI: 10.1103/PhysRevLett.69.2863Google Scholar1https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2sfptF2isg%253D%253D&md5=51e8562b250f575cd902524cde61c5d1Density matrix formulation for quantum renormalization groupsWhitePhysical review letters (1992), 69 (19), 2863-2866 ISSN:.There is no expanded citation for this reference.**2**White, S. R.; Martin, R. L. Ab initio quantum chemistry using the density matrix renormalization group.*J. Chem. Phys.*1999,*110*, 4127– 4130, DOI: 10.1063/1.478295Google Scholar2https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXhtF2gtbw%253D&md5=ae0c47542a0ddc08171b93f29693e51fAb initio quantum chemistry using the density matrix renormalization groupWhite, Steven R.; Martin, Richard L.Journal of Chemical Physics (1999), 110 (9), 4127-4130CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper we describe how the d. matrix renormalization group can be used for quantum chem. calcns. for mols., as an alternative to traditional methods, such as CI or coupled cluster approaches. As a demonstration of the potential of this approach, we present results for the H2O mol. in a std. Gaussian basis. Results for the total energy of the system compare favorably with the best traditional quantum chem. methods.**3**Chan, G. K.-L.; Head-Gordon, M. Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group.*J. Chem. Phys.*2002,*116*, 4462– 4476, DOI: 10.1063/1.1449459Google Scholar3https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhslCjsrs%253D&md5=f1249212b5c6414a901ff5c8a4a64542Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization groupChan, Garnet Kin-Lic; Head-Gordon, MartinJournal of Chemical Physics (2002), 116 (11), 4462-4476CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the recently developed D. Matrix Renormalization Group (DMRG) algorithm in the context of quantum chem. In contrast to traditional approaches, this algorithm is believed to yield arbitrarily high accuracy in the energy with only polynomial computational effort. We describe in some detail how this is achieved. We begin by introducing the principles of the renormalization procedure, and how one formulates an algorithm for use in quantum chem. The renormalization group algorithm is then interpreted in terms of familiar quantum chem. concepts, and its numerical behavior, including its convergence and computational cost, are studied using both model and real systems. The asymptotic convergence of the algorithm is derived. Finally, we examine the performance of the DMRG on widely studied chem. problems, such as the water mol., the twisting barrier of ethene, and the dissocn. of nitrogen. In all cases, the results are favorably comparable with the best existing quantum chem. methods, and particularly so when the nondynamical correlation is strong. Some perspectives for future development are given.**4**Legeza, Ö.; Röder, J.; Hess, B. A. Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach.*Phys. Rev. B*2003,*67*, 125114, DOI: 10.1103/PhysRevB.67.125114Google Scholar4https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXivFaqsbY%253D&md5=0f5650c8557f55974dea14ef3e8a41d3Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approachLegeza, O.; Roder, J.; Hess, B. A.Physical Review B: Condensed Matter and Materials Physics (2003), 67 (12), 125114/1-125114/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)We have applied the momentum space version of the d.-matrix renormalization-group method (k-DMRG) in quantum chem. in order to study the accuracy of the algorithm in this new context. We have shown numerically that it is possible to det. the desired accuracy of the method in advance of the calcns. by dynamically controlling the truncation error and the no. of block states using a novel protocol that we dubbed dynamical block state selection protocol. The relationship between the real error and truncation error has been studied as a function of the no. of orbitals and the fraction of filled orbitals. We have calcd. the ground state of the mols. CH2, H2O, and F2 as well as the first excited state of CH2. Our largest calcns. were carried out with 57 orbitals, the largest no. of block states was 1500-2000, and the largest dimensions of the Hilbert space of the superblock configuration was 800 000-1 200 000.**5**Schollwöck, U. The density-matrix renormalization group in the age of matrix product states.*Annals of Physics*2011,*326*, 96– 192, DOI: 10.1016/j.aop.2010.09.012Google ScholarThere is no corresponding record for this reference.**6**Olivares-Amaya, R.; Hu, W.; Nakatani, N.; Sharma, S.; Yang, J.; Chan, G. K.-L. The ab-initio density matrix renormalization group in practice.*J. Chem. Phys.*2015,*142*, 034102, DOI: 10.1063/1.4905329Google Scholar6https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXovFSmsQ%253D%253D&md5=4e30986e7c45a42b1df2a78031f17c58The ab-initio density matrix renormalization group in practiceOlivares-Amaya, Roberto; Hu, Weifeng; Nakatani, Naoki; Sharma, Sandeep; Yang, Jun; Chan, Garnet Kin-LicJournal of Chemical Physics (2015), 142 (3), 034102/1-034102/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The ab-initio d. matrix renormalization group (DMRG) is a tool that can be applied to a wide variety of interesting problems in quantum chem. Here, we examine the d. matrix renormalization group from the vantage point of the quantum chem. user. What kinds of problems is the DMRG well-suited to. What are the largest systems that can be treated at practical cost. What sort of accuracies can be obtained, and how do we reason about the computational difficulty in different mols.. By examg. a diverse benchmark set of mols.: π-electron systems, benchmark main-group and transition metal dimers, and the Mn-oxo-salen and Fe-porphine organometallic compds., we provide some answers to these questions, and show how the d. matrix renormalization group is used in practice. (c) 2015 American Institute of Physics.**7**Kurashige, Y.; Yanai, T. Second-order perturbation theory with a density matrix renormalization group self-consistent field reference function: Theory and application to the study of chromium dimer.*J. Chem. Phys.*2011,*135*, 094104, DOI: 10.1063/1.3629454Google Scholar7https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtFajtr%252FF&md5=3555ac1743fc963a44b9c86c08168517Second-order perturbation theory with a density matrix renormalization group self-consistent field reference function: Theory and application to the study of chromium dimerKurashige, Yuki; Yanai, TakeshiJournal of Chemical Physics (2011), 135 (9), 094104/1-094104/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a second-order perturbation theory based on a d. matrix renormalization group SCF (DMRG-SCF) ref. function. The method reproduces the soln. of the complete active space with second-order perturbation theory (CASPT2) when the DMRG ref. function is represented by a sufficiently large no. of renormalized many-body basis, thereby being named DMRG-CASPT2 method. The DMRG-SCF is able to describe non-dynamical correlation with large active space that is insurmountable to the conventional CASSCF method, while the second-order perturbation theory provides an efficient description of dynamical correlation effects. The capability of our implementation is demonstrated for an application to the potential energy curve of the chromium dimer, which is one of the most demanding multireference systems that require best electronic structure treatment for non-dynamical and dynamical correlation as well as large basis sets. The DMRG-CASPT2/cc-pwCV5Z calcns. were performed with a large (3d double-shell) active space consisting of 28 orbitals. Our approach using large-size DMRG ref. addressed the problems of why the dissocn. energy is largely overestimated by CASPT2 with the small active space consisting of 12 orbitals (3d4s), and also is oversensitive to the choice of the zeroth-order Hamiltonian. (c) 2011 American Institute of Physics.**8**Saitow, M.; Kurashige, Y.; Yanai, T. Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave function.*J. Chem. Phys.*2013,*139*, 044118, DOI: 10.1063/1.4816627Google Scholar8https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhtFygsbnL&md5=a30f97f373ba1a069bbce1c23a48e178Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave functionSaitow, Masaaki; Kurashige, Yuki; Yanai, TakeshiJournal of Chemical Physics (2013), 139 (4), 044118/1-044118/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We report development of the multireference CI (MRCI) method that can use active space scalable to much larger size refs. than has previously been possible. The recent development of the d. matrix renormalization group (DMRG) method in multireference quantum chem. offers the ability to describe static correlation in a large active space. The present MRCI method provides a crit. correction to the DMRG ref. by including high-level dynamic correlation through the CI treatment. When the DMRG and MRCI theories are combined (DMRG-MRCI), the full internal contraction of the ref. in the MRCI ansatz, including contraction of semi-internal states, plays a central role. However, it is thought to involve formidable complexity because of the presence of the five-particle rank reduced-d. matrix (RDM) in the Hamiltonian matrix elements. To address this complexity, we express the Hamiltonian matrix using commutators, which allows the five-particle rank RDM to be canceled out without any approxn. Then we introduce an approxn. to the four-particle rank RDM by using a cumulant reconstruction from lower-particle rank RDMs. A computer-aided approach is employed to derive the exceedingly complex equations of the MRCI in tensor-contracted form and to implement them into an efficient parallel computer code. This approach extends to the size-consistency-cor. variants of MRCI, such as the MRCI+Q, MR-ACPF, and MR-AQCC methods. We demonstrate the capability of the DMRG-MRCI method in several benchmark applications, including the evaluation of single-triplet gap of free-base porphyrin using 24 active orbitals. (c) 2013 American Institute of Physics.**9**Wouters, S.; Nakatani, N.; Van Neck, D.; Chan, G. K.-L. Thouless theorem for matrix product states and subsequent post density matrix renormalization group methods.*Phys. Rev. B*2013,*88*, 075122, DOI: 10.1103/PhysRevB.88.075122Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhsFykt7fN&md5=d0c396e966684bcc99590f31a9d885e6Thouless theorem for matrix product states and subsequent post density matrix renormalization group methodsWouters, Sebastian; Nakatani, Naoki; Van Neck, Dimitri; Chan, Garnet Kin-LicPhysical Review B: Condensed Matter and Materials Physics (2013), 88 (7), 075122/1-075122/14CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)The similarities between Hartree-Fock (HF) theory and the d. matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function Ansatz. Linearization of the time-dependent variational principle near a variational min. allows to derive the RPA (RPA). We show that the nonredundant parameterization of the matrix product state (MPS) tangent space leads to the Thouless theorem for MPS, i.e., an explicit nonredundant parameterization of the entire MPS manifold, starting from a specific MPS ref. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approxn. (TDA), the CI (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS Ansatz with single and double excitations to improve on the ground state and to calc. low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calcns. of the RPA-MPS correlation energy. With the long-range quantum chem. PPP Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.**10**Yanai, T.; Chan, G. K.-L. Canonical transformation theory for multireference problems.*J. Chem. Phys.*2006,*124*, 194106, DOI: 10.1063/1.2196410Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XltVOjsbs%253D&md5=15d27491b9be26ab2fe5888182f60b48Canonical transformation theory for multireference problemsYanai, Takeshi; Chan, Garnet Kin-LicJournal of Chemical Physics (2006), 124 (19), 194106/1-194106/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We propose a theory to describe dynamic correlations in bonding situations where there is also significant nondynamic character. We call this the canonical transformation (CT) theory. When combined with a suitable description of nondynamic correlation, such as given by a complete-active-space SCF (CASSCF) or d. matrix renormalization group wave function, it provides a theory to describe bonding situations across the entire potential energy surface with quant. accuracy for both dynamic and nondynamic correlation. The canonical transformation theory uses a unitary exponential ansatz, is size consistent, and has a computational cost of the same order as a single-ref. coupled cluster theory with the same level of excitations. Calcns. using the CASSCF based CT method with single and double operators for the potential energy curves for water and nitrogen mols., the BeH2 insertion reaction, and hydrogen fluoride and boron hydride bond breaking, consistently yield quant. accuracies typical of equil. region coupled cluster theory, but across all geometries, and better than obtained with multireference perturbation theory.**11**Ren, J.; Yi, Y.; Shuai, Z. Inner Space Perturbation Theory in Matrix Product States: Replacing Expensive Iterative Diagonalization.*J. Chem. Theory Comput.*2016,*12*, 4871– 4878, DOI: 10.1021/acs.jctc.6b00696Google Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsVKrt7nM&md5=999a0e303af01c951094be2ef7b64ee6Inner Space Perturbation Theory in Matrix Product States: Replacing Expensive Iterative DiagonalizationRen, Jiajun; Yi, Yuanping; Shuai, ZhigangJournal of Chemical Theory and Computation (2016), 12 (10), 4871-4878CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We propose an inner space perturbation theory (isPT) to replace the expensive iterative diagonalization in the std. d. matrix renormalization group theory (DMRG). The retained reduced d. matrix eigenstates are partitioned into the active and secondary space. The first-order wave function and the second- and third-order energies are easily computed by using one step Davidson iteration. Our formulation has several advantages including (i) keeping a balance between the efficiency and accuracy, (ii) capturing more entanglement with the same amt. of computational time, (iii) recovery of the std. DMRG when all the basis states belong to the active space. Numerical examples for the polyacenes and periacene show that the efficiency gain is considerable and the accuracy loss due to the perturbation treatment is very small, when half of the total basis states belong to the active space. Moreover, the perturbation calcns. converge in all our numerical examples.**12**Beran, P.; Matoušek, M.; Hapka, M.; Pernal, K.; Veis, L. Density matrix renormalization group with dynamical correlation via adiabatic connection.*J. Chem. Theory Comput.*2021,*17*, 7575, DOI: 10.1021/acs.jctc.1c00896Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXisVGrsLjL&md5=6dfdecdd844b5a539f1bd28e0d77e612Density Matrix Renormalization Group with Dynamical Correlation via Adiabatic ConnectionBeran, Pavel; Matousek, Mikulas; Hapka, Michal; Pernal, Katarzyna; Veis, LiborJournal of Chemical Theory and Computation (2021), 17 (12), 7575-7585CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The quantum chem. version of the d. matrix renormalization group (DMRG) method has established itself as one of the methods of choice for calcns. of strongly correlated mol. systems. Despite its great ability to capture strong electronic correlation in large active spaces, it is not suitable for computations of dynamical electron correlation. In this work, we present a new approach to the electronic structure problem of strongly correlated mols., in which DMRG is responsible for a proper description of the strong correlation, whereas dynamical correlation is computed via the recently developed adiabatic connection (AC) technique which requires only up to two-body active space reduced d. matrixes. We report the encouraging results of this approach on typical candidates for DMRG computations, namely, n-acenes (n = 2 → 7), Fe(II)-porphyrin, and the Fe3S4 cluster.**13**Barcza, G.; Werner, M. A.; Zaránd, G.; Pershin, A.; Benedek, Z.; Legeza, Ö.; Szilvási, T. Toward Large-Scale Restricted Active Space Calculations Inspired by the Schmidt Decomposition.*J. Phys. Chem. A*2022,*126*, 9709– 9718, DOI: 10.1021/acs.jpca.2c05952Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XjtFSht7nO&md5=5f4a7788606b693ab9dca3e86f4e1314Toward large-scale restricted active space calculations inspired by the Schmidt decompositionBarcza, Gergely; Werner, Miklos Antal; Zarand, Gergely; Pershin, Anton; Benedek, Zsolt; Legeza, Ors; Szilvasi, TiborJournal of Physical Chemistry A (2022), 126 (51), 9709-9718CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present an alternative, memory-efficient, Schmidt decompn.-based description of the inherently bipartite restricted active space (RAS) scheme, which can be implemented effortlessly within the d. matrix renormalization group (DMRG) method via the dynamically extended active space procedure. Benchmark calcns. are compared against state-of-the-art results of C2 and Cr2, which are notorious for their multireference character. Our results for ground and excited states together with spectroscopic consts. demonstrate that the proposed novel approach, dubbed as DMRG-RAS, which is variational and free of uncontrolled method errors, has the potential to outperfom conventional methods for strongly correlated mols.**14**Friesecke, G.; Barcza, G.; Legeza, O. Predicting the FCI energy of large systems to chemical accuracy from restricted active space density matrix renormalization group calculations.*J. Chem. Theory Comput.*2024,*20*, 87– 102, DOI: 10.1021/acs.jctc.3c01001Google ScholarThere is no corresponding record for this reference.**15**Veis, L.; Antalík, A.; Brabec, J.; Neese, F.; Legeza, Ö.; Pittner, J. Coupled Cluster Method with Single and Double Excitations Tailored by Matrix Product State Wave Functions.*J. Phys. Chem. Lett.*2016,*7*, 4072– 4078, DOI: 10.1021/acs.jpclett.6b01908Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsFOktrbL&md5=6f0a8d79fff88c257aef0327abd74296Coupled Cluster Method with Single and Double Excitations Tailored by Matrix Product State Wave FunctionsVeis, Libor; Antalik, Andrej; Brabec, Jiri; Neese, Frank; Legeza, Ors; Pittner, JiriJournal of Physical Chemistry Letters (2016), 7 (20), 4072-4078CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)In the past decade, the quantum chem. version of the d. matrix renormalization group (DMRG) method has established itself as the method of choice for calcns. of strongly correlated mol. systems. Despite its favorable scaling, it is in practice not suitable for computations of dynamic correlation. We present a novel method for accurate "post-DMRG" treatment of dynamic correlation based on the tailored coupled cluster (CC) theory in which the DMRG method is responsible for the proper description of nondynamic correlation, whereas dynamic correlation is incorporated through the framework of the CC theory. We illustrate the potential of this method on prominent multireference systems, in particular, N2 and Cr2 mols. and also oxo-Mn(Salen), for which we have performed the first post-DMRG computations in order to shed light on the energy ordering of the lowest spin states.**16**Kinoshita, T.; Hino, O.; Bartlett, R. J. Coupled-cluster method tailored by configuration interaction.*J. Chem. Phys.*2005,*123*, 074106, DOI: 10.1063/1.2000251Google Scholar16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpsleht7w%253D&md5=7c005bedda722c576125ea4cec764785Coupled-cluster method tailored by configuration interactionKinoshita, Tomoko; Hino, Osamu; Bartlett, Rodney J.Journal of Chemical Physics (2005), 123 (7), 074106/1-074106/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A method is presented which combines coupled cluster (CC) and CI to describe accurately potential-energy surfaces (PESs). We use the cluster amplitudes extd. from the complete active space CI calcn. to manipulate nondynamic correlation to tailor a single ref. CC theory (TCC). The dynamic correlation is then incorporated through the framework of the CC method. We illustrate the method by describing the PESs for HF, H2O, and N2 mols. which involve single, double, and triple bond-breaking processes. To the dissocn. limit, this approach yields far more accurate PESs than those obtained from the conventional CC method and the addnl. computational cost is negligible compared with the CC calcn. steps. We anticipate that TCC offers an effective and generally applicable approach for many problems.**17**Hino, O.; Kinoshita, T.; Chan, G. K.-L.; Bartlett, R. J. Tailored coupled cluster singles and doubles method applied to calculations on molecular structure and harmonic vibrational frequencies of ozone.*J. Chem. Phys.*2006,*124*, 114311, DOI: 10.1063/1.2180775Google Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XivV2jtrs%253D&md5=57806470896d5932ad362d871712636dTailored coupled cluster singles and doubles method applied to calculations on molecular structure and harmonic vibrational frequencies of ozoneHino, Osamu; Kinoshita, Tomoko; Chan, Garnet Kin-Lic; Bartlett, Rodney J.Journal of Chemical Physics (2006), 124 (11), 114311/1-114311/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)To assess the sepn. of dynamic and nondynamic correlations and orbital choice, we calc. the mol. structure and harmonic vibrational frequencies of ozone with the recently developed tailored coupled cluster singles and doubles method (TCCSD). We employ the Hartree-Fock and complete active space (CAS) SCF orbitals to perform TCCSD calcns. When using the Hartree-Fock orbitals, it is difficult to reproduce the exptl. vibrational frequency of the asym. stretching mode. On the other hand, the TCCSD based on the CASSCF orbitals in a correlation consistent polarized valence triple zeta basis yields excellent results with the two sym. vibrations differing from the exptl. harmonic values by 2 cm-1 and the asym. vibration differing by 9 cm-1.**18**Veis, L.; Antalík, A.; Brabec, J.; Neese, F.; Legeza, Ö.; Pittner, J.*J. Phys. Chem. Lett.*2017,*8*, 291, DOI: 10.1021/acs.jpclett.6b02912Google ScholarThere is no corresponding record for this reference.**19**Faulstich, F. M.; Laestadius, A.; Legeza, Ö.; Schneider, R.; Kvaal, S. Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry.*SIAM Journal on Numerical Analysis*2019,*57*, 2579– 2607, DOI: 10.1137/18M1171436Google ScholarThere is no corresponding record for this reference.**20**Faulstich, F. M.; Máté, M.; Laestadius, A.; Csirik, M. A.; Veis, L.; Antalik, A.; Brabec, J.; Schneider, R.; Pittner, J.; Kvaal, S.; Legeza, Ö. Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer.*J. Chem. Theory Comput.*2019,*15*, 2206– 2220, DOI: 10.1021/acs.jctc.8b00960Google Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXjsleju7c%253D&md5=a1b71dd62855082fb746af6f6fbcb493Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen DimerFaulstich, Fabian M.; Mate, Mihaly; Laestadius, Andre; Csirik, Mihaly Andras; Veis, Libor; Antalik, Andrej; Brabec, Jiri; Schneider, Reinhold; Pittner, Jiri; Kvaal, Simen; Legeza, OrsJournal of Chemical Theory and Computation (2019), 15 (4), 2206-2220CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In this article, we investigate the numerical and theor. aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing math. anal. is here elaborated in a quantum chem. framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO-LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the d. matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the d. matrix renormalization group-tailored coupled-cluster error.**21**Leszczyk, A.; Máté, M.; Legeza, Ö.; Boguslawski, K. Assessing the accuracy of tailored coupled cluster methods corrected by electronic wave functions of polynomial cost.*J. Chem. Theory Comp.*2022,*18*, 96, DOI: 10.1021/acs.jctc.1c00284Google ScholarThere is no corresponding record for this reference.**22**Neese, F. The ORCA program system.*WIREs Computational Molecular Science*2012,*2*, 73– 78, DOI: 10.1002/wcms.81Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvFGls7s%253D&md5=a753e33a6f9a326553295596f5c754e5The ORCA program systemNeese, FrankWiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (1), 73-78CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. ORCA is a general-purpose quantum chem. program package that features virtually all modern electronic structure methods (d. functional theory, many-body perturbation and coupled cluster theories, and multireference and semiempirical methods). It is designed with the aim of generality, extendibility, efficiency, and user friendliness. Its main field of application is larger mols., transition metal complexes, and their spectroscopic properties. ORCA uses std. Gaussian basis functions and is fully parallelized. The article provides an overview of its current possibilities and documents its efficiency.**23**Antalik, A.; Veis, L.; Brabec, J.; Demel, O.; Legeza, O.; Pittner, J. Toward the efficient local tailored coupled cluster approximation and the peculiar case of oxo-Mn(Salen).*J. Chem. Phys.*2019,*151*, 084112, DOI: 10.1063/1.5110477Google ScholarThere is no corresponding record for this reference.**24**Lang, J.; Antalik, A.; Veis, L.; Brandejs, J.; Brabec, J.; Legeza, O.; Pittner, J. Near-linear Scaling in DMRG-based Tailored Coupled Clusters: An Implementation of DLPNO-TCCSD and DLPNO-TCCSD(T).*J. Chem. Theor. Comput.*2020,*16*, 3028, DOI: 10.1021/acs.jctc.0c00065Google ScholarThere is no corresponding record for this reference.**25**Antalik, A.; Nachtigallova, D.; Lo, R.; Matousek, M.; LAng, J.; Legeza, O.; Pittner, J.; Hobza, P.; Veis, L. Ground state of the Fe(II)-porphyrin model system corresponds to the quintet state: DFT, DMRG-TCCSD and DMRG-TCCSD(T) computations.*Phys. Chem. Chem. Phys.*2020,*22*, 17033, DOI: 10.1039/D0CP03086DGoogle ScholarThere is no corresponding record for this reference.**26**Brandejs, J.; Višňák, J.; Veis, L.; Maté, M.; Legeza, O.; Pittner, J. Toward DMRG-tailored coupled cluster method in the 4c-relativistic domain.*J. Chem. Phys.*2020,*152*, 174107, DOI: 10.1063/1.5144974Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXovFGktr8%253D&md5=0d571efd9ea26884e1055dcd374a4918Toward DMRG-tailored coupled cluster method in the 4c-relativistic domainBrandejs, Jan; Visnak, Jakub; Veis, Libor; Mate, Mihaly; Legeza, Ors; Pittner, JiriJournal of Chemical Physics (2020), 152 (17), 174107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)There are three essential problems in computational relativistic chem.: Electrons moving at relativistic speeds, close lying states, and dynamical correlation. Currently available quantum-chem. methods are capable of solving systems with one or two of these issues. However, there is a significant class of mols. in which all the three effects are present. These are the heavier transition metal compds., lanthanides, and actinides with open d or f shells. For such systems, sufficiently accurate numerical methods are not available, which hinders the application of theor. chem. in this field. In this paper, we combine two numerical methods in order to address this challenging class of mols. These are the relativistic versions of coupled cluster methods and the d. matrix renormalization group (DMRG) method. To the best of our knowledge, this is the first relativistic implementation of the coupled cluster method externally cor. by DMRG. The method brings a significant redn. of computational costs as we demonstrate on the system of TlH, AsH, and SbH. (c) 2020 American Institute of Physics.**27**Wormit, M.; Olejniczak, M.łg.; Deppenmeier, A.-L.; Borschevsky, A.; Saue, T.; Schwerdtfeger, P. Strong enhancement of parity violation effects in chiral uranium compounds.*Phys. Chem. Chem. Phys.*2014,*16*, 17043– 17051, DOI: 10.1039/C4CP01904KGoogle Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVKgsr7L&md5=dddf4e351522977f2dd9a619151d4a44Strong enhancement of parity violation effects in chiral uranium compoundsWormit, Michael; Olejniczak, Malgorzata; Deppenmeier, Anna-Lena; Borschevsky, Anastasia; Saue, Trond; Schwerdtfeger, PeterPhysical Chemistry Chemical Physics (2014), 16 (32), 17043-17051CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)The effects of parity violation (PV) on the vibrational transitions of chiral uranium compds. of the type N≡UXYZ and N≡UHXY (X, Y, Z = F, Cl, Br, I) are analyzed by means of exact two-component relativistic (X2C) Hartree-Fock and d. functional calcns. using NUFClI and NUHFI as representative examples. The PV contributions to the vibrational transitions were found to be in the Hz range, larger than for any of the earlier proposed chiral mols. Thus, these systems are very promising candidates for future exptl. PV measurements. A detailed comparison of the N≡UHFI and the N≡WHFI homologues reveals that subtle electronic structure effects, rather than exclusively a simple Z5 scaling law, are the cause of the strong enhancement in PV contributions of the chiral uranium mols.**28**Jiang, W.; Wilson, A. K. Multireference composite approaches for the accurate study of ground and excited electronic states: C2, N2, and O2.*J. Chem. Phys.*2011,*134*, 034101, DOI: 10.1063/1.3514031Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmvFWrtQ%253D%253D&md5=d1460e6260a528382a9fbfd151f2d1d5Multireference composite approaches for the accurate study of ground and excited electronic states: C2, N2, and O2Jiang, Wanyi; Wilson, Angela K.Journal of Chemical Physics (2011), 134 (3), 034101/1-034101/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A multiref. analog of the correlation consistent composite approach (MR-ccCA) based on complete active space with 2nd-order perturbation theory (CASPT2) was used in a study of the ground and valence excited states of C2, N2, and O2. The performance of different 2nd-order multiref. perturbation theory methods including 2nd-order n-electron valence state perturbation theory, 2nd-order multiref. Moller-Plesset, and 2nd-order generalized van Vleck perturbation theory was analyzed as potential alternatives to CASPT2 within MR-ccCA. The MR-ccCA-P predicts spectroscopic consts. with overall mean abs. deviations from exptl. values of 0.0006 Å, 7.0 cm-1, and 143 cm-1 for equil. bond length (re), harmonic frequency (ωe), and term values (Te), resp., which are comparable to the predictions by more computationally costly multiref. CI-based methods. (c) 2011 American Institute of Physics.**29**Mintz, B.; Williams, T. G.; Howard, L.; Wilson, A. K. Computation of potential energy surfaces with the multireference correlation consistent composite approach.*J. Chem. Phys.*2009,*130*, 234104, DOI: 10.1063/1.3149387Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXnsFClurw%253D&md5=e336a765219902a7b85523063806534cComputation of potential energy surfaces with the multireference correlation consistent composite approachMintz, Benjamin; Williams, T. Gavin; Howard, Levi; Wilson, Angela K.Journal of Chemical Physics (2009), 130 (23), 234104/1-234104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A multireference composite method that is based on the correlation consistent Composite Approach (ccCA) is introduced. The developed approach, multireference ccCA, was utilized to compute the potential energy surfaces (PESs) of N2 and C2, which provide rigorous tests for multireference composite methods due to the large multireference character that must be correctly described as the mols. dissoc. As well, PESs provide a stringent test of a composite method because all components of the method must work in harmony for an appropriate, smooth representation across the entire surface. (c) 2009 American Institute of Physics.**30**Chan, G. K.-L.; Kállay, M.; Gauss, J. State-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curve.*J. Chem. Phys.*2004,*121*, 6110– 6116, DOI: 10.1063/1.1783212Google Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXnvFeqs7g%253D&md5=409eb74e4f9716d5bab8ddf7de1e4f8bState-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curveChan, Garnet Kin-Lic; Kallay, Mihaly; Gauss, JurgenJournal of Chemical Physics (2004), 121 (13), 6110-6116CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the nitrogen binding curve with the d. matrix renormalization group (DMRG) and single-ref. and multireference coupled cluster (CC) theory. Our DMRG calcns. use up to 4000 states and our single-ref. CC calcns. include up to full connected hextuple excitations. Using the DMRG, we compute an all-electron benchmark nitrogen binding curve, at the polarized, valence double-zeta level (28 basis functions), with an estd. accuracy of 0.03 mEh. We also assess the performance of more approx. DMRG and CC theories across the nitrogen curve. We provide an anal. of the relative strengths and merits of the DMRG and CC theory under different correlation conditions.**31**Máté, M.; Petrov, K.; Szalay, S.; Legeza, Ö. Compressing multireference character of wave functions via fermionic mode optimization.*J. Math. Chem.*2023,*61*, 362– 375, DOI: 10.1007/s10910-022-01379-yGoogle Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XhvFOgs7vJ&md5=ce681e458caee0bd135ea33dd7d87c2cCompressing multireference character of wave functions via fermionic mode optimizationMate, Mihaly; Petrov, Klara; Szalay, Szilard; Legeza, OrsJournal of Mathematical Chemistry (2023), 61 (2), 362-375CODEN: JMCHEG; ISSN:0259-9791. (Springer)Abstr.: In this work, we present a brief overview of the fermionic mode optimization within the framework of tensor network state methods (Krumnow et al. in Phys Rev Lett 117:210402, 2016, https://doi.org/10.1103/PhysRevLett.117.210402), and demonstrate that it has the potential to compress the multireference character of the wave functions after finding optimal MOs (modes), based on entanglement minimization. Numerical simulations have been performed for the nitrogen dimer in the cc-pVDZ basis for the equil. and for stretched geometries.**32**Boguslawski, K.; Tecmer, P. Benchmark of Dynamic Electron Correlation Models for Seniority-Zero Wave Functions and Their Application to Thermochemistry.*J. Chem. Theory Comput.*2017,*13*, 5966– 5983, DOI: 10.1021/acs.jctc.6b01134Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFWgsrzI&md5=aa12162e85f9aaae838d5117a2ace7d7Benchmark of Dynamic Electron Correlation Models for Seniority-Zero Wave Functions and Their Application to ThermochemistryBoguslawski, Katharina; Tecmer, PawelJournal of Chemical Theory and Computation (2017), 13 (12), 5966-5983CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Wave functions restricted to electron-pair states are promising models to describe static/nondynamic electron correlation effects encountered, for instance, in bond-dissocn. processes and transition-metal and actinide chem. To reach spectroscopic accuracy, however, the missing dynamic electron correlation effects that cannot be described by electron-pair states need to be included a posteriori. In this Article, we extend the previously presented perturbation theory models with an Antisym. Product of 1-ref. orbital Geminal (AP1roG) ref. function that allows us to describe both static/nondynamic and dynamic electron correlation effects. Specifically, our perturbation theory models combine a diagonal and off-diagonal zero-order Hamiltonian, a single-ref. and multireference dual state, and different excitation operators used to construct the projection manifold. We benchmark all proposed models as well as an a posteriori Linearized Coupled Cluster correction on top of AP1roG against CR-CC(2,3) ref. data for reaction energies of several closed-shell mols. that are extrapolated to the basis set limit. Moreover, we test the performance of our new methods for multiple bond breaking processes in the homonuclear N2, C2, and F2 dimers as well as the heteronuclear BN, CO, and CN+ dimers against MRCI-SD, MRCI-SD + Q, and CR-CC(2,3) ref. data. Our numerical results indicate that the best performance is obtained from a Linearized Coupled Cluster correction as well as second-order perturbation theory corrections employing a diagonal and off-diagonal zero-order Hamiltonian and a single-determinant dual state. These dynamic corrections on top of AP1roG provide substantial improvements for binding energies and spectroscopic properties obtained with the AP1roG approach, while allowing us to approach chem. accuracy for reaction energies involving closed-shell species.PMID: 28921980.

**33**Nowak, A.; Legeza, O.; Boguslawski, K. Orbital entanglement and correlation from pCCD-tailored coupled cluster wave functions.*J. Chem. Phys.*2021,*154*, 084111, DOI: 10.1063/5.0038205Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXltFOntL0%253D&md5=e935016c0969f409fbf7ec34903598d7Orbital entanglement and correlation from pCCD-tailored coupled cluster wave functionsNowak, Artur; Legeza, Ors; Boguslawski, KatharinaJournal of Chemical Physics (2021), 154 (8), 084111CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Wave functions based on electron-pair states provide inexpensive and reliable models to describe quantum many-body problems contg. strongly correlated electrons, given that broken-pair states have been appropriately accounted for by, for instance, a posteriori corrections. In this article, we analyze the performance of electron-pair methods in predicting orbital-based correlation spectra. We focus on the (orbital-optimized) pair-coupled cluster doubles (pCCD) ansatz with a linearized coupled-cluster (LCC) correction. Specifically, we scrutinize how orbital-based entanglement and correlation measures can be detd. from a pCCD-tailored CC wave function. Furthermore, we employ the single-orbital entropy, the orbital-pair mutual information, and the eigenvalue spectra of the two-orbital reduced d. matrixes to benchmark the performance of the LCC correction for the one-dimensional Hubbard model with the periodic boundary condition as well as the N2 and F2 mols. against d. matrix renormalization group ref. calcns. Our study indicates that pCCD-LCC accurately reproduces the orbital-pair correlation patterns in the weak correlation limit and for mols. close to their equil. structure. Hence, we can conclude that pCCD-LCC predicts reliable wave functions in this regime. (c) 2021 American Institute of Physics.**34**Leszczyk, A.; Máté, M.; Legeza, O.; Boguslawski, K. Assessing the Accuracy of Tailored Coupled Cluster Methods Corrected by Electronic Wave Functions of Polynomial Cost.*J. Chem. Theory Comput.*2022,*18*, 96– 117, DOI: 10.1021/acs.jctc.1c00284Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXivVeksrvN&md5=5785b5dbc6bb45629aa9eb30c5a00797Assessing the Accuracy of Tailored Coupled Cluster Methods Corrected by Electronic Wave Functions of Polynomial CostLeszczyk, Aleksandra; Mate, Mihaly; Legeza, Ors; Boguslawski, KatharinaJournal of Chemical Theory and Computation (2022), 18 (1), 96-117CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Tailored coupled cluster theory represents a computationally inexpensive way to describe static and dynamical electron correlation effects. In this work, we scrutinize the performance of various coupled cluster methods tailored by electronic wave functions of polynomial cost. Specifically, we focus on frozen-pair coupled cluster (fpCC) methods, which are tailored by pair-coupled cluster doubles (pCCD), and coupled cluster theory tailored by matrix product state wave functions optimized by the d. matrix renormalization group (DMRG) algorithm. As test system, we selected a set of various small- and medium-sized mols. contg. diatomics (N2, F2, C2, CN+, CO, BN, BO+, and Cr2) and mols. (ammonia, ethylene, cyclobutadiene, benzene, hydrogen chains, rings, and cuboids) for which the conventional single-ref. coupled cluster singles and doubles (CCSD) method is not able to produce accurate results for spectroscopic consts., potential energy surfaces, and barrier heights. Most importantly, DMRG-tailored and pCCD-tailored approaches yield similar errors in spectroscopic consts. and potential energy surfaces compared to accurate theor. and/or exptl. ref. data. Although fpCC methods provide a reliable description for the dissocn. pathway of mols. featuring single and quadruple bonds, they fail in the description of triple or hextuple bond-breaking processes or avoided crossing regions.**35**Nowak, A.; Boguslawski, K. A configuration interaction correction on top of pair coupled cluster doubles.*Phys. Chem. Chem. Phys.*2023,*25*, 7289– 7301, DOI: 10.1039/D2CP05171KGoogle Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3sXjs1emsLo%253D&md5=a82af0b8ba3799cb795584540a76a282A configuration interaction correction on top of pair coupled cluster doublesNowak, Artur; Boguslawski, KatharinaPhysical Chemistry Chemical Physics (2023), 25 (10), 7289-7301CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Numerous numerical studies have shown that geminal-based methods are a promising direction to model strongly correlated systems with low computational costs. Several strategies have been introduced to capture the missing dynamical correlation effects, which typically exploit a posteriori corrections to account for correlation effects assocd. with broken-pair states or inter-geminal correlations. In this article, we scrutinize the accuracy of the pair coupled cluster doubles (pCCD) method extended by CI (CI) theory. Specifically, we benchmark various CI models, including, at most double excitations against selected CC corrections as well as conventional single-ref. CC methods. A simple Davidson correction is also tested. The accuracy of the proposed pCCD-CI approaches is assessed for challenging small model systems such as the N2 and F2 dimers and various di- and triat. actinide-contg. compds. In general, the proposed CI methods considerably improve spectroscopic consts. compared to the conventional CCSD approach, provided a Davidson correction is included in the theor. model. At the same time, their accuracy lies between those of the linearized frozen pCCD and frozen pCCD variants.**36**Li, X.; Paldus, J. Dissociation of N2 triple bond: a reduced multireference CCSD study.*Chem. Phys. Lett.*1998,*286*, 145– 154, DOI: 10.1016/S0009-2614(97)01132-9Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXhvF2isLY%253D&md5=1903dac9e4922e561f848ffc7f2472daDissociation of N2 triple bond: a reduced multireference CCSD studyLi, Xiangzhu; Paldus, JosefChemical Physics Letters (1998), 286 (1,2), 145-154CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)The reduced multireference coupled cluster method with singles and doubles is applied to the dissocn. of the ground state of the N mol. Even with a relatively modest highly truncated ref. space one obtains an accurate potential over a wide range of internuclear sepns.**37**Leszczyk, A.; Dome, T.; Tecmer, P.; Kedziera, D.; Boguslawski, K. Resolving the π-assisted U–N σf-bond formation using quantum information theory.*Phys. Chem. Chem. Phys.*2022,*24*, 21296– 21307, DOI: 10.1039/D2CP03377AGoogle ScholarThere is no corresponding record for this reference.**38**Kramers, H. A. Théorie générale de la rotation paramagnétique dans les cristaux.*Proceedings of the Royal Netherlands Academy of Arts and Sciences*1930,*33*(6-10), 959– 972Google ScholarThere is no corresponding record for this reference.**39**Fleig, T.; Olsen, J.; Marian, C. M. The generalized active space concept for the relativistic treatment of electron correlation. I. Kramers-restricted two-component configuration interaction.*J. Chem. Phys.*2001,*114*, 4775– 4790, DOI: 10.1063/1.1349076Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXhslWisLo%253D&md5=406052d6433f64aec91b79c33cd878a8The generalized active space concept for the relativistic treatment of electron correlation. I. Kramers-restricted two-component configuration interactionFleig, Timo; Olsen, Jeppe; Marian, Christel M.Journal of Chemical Physics (2001), 114 (11), 4775-4790CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)As a prelude to a series of presentations dealing with the treatment of electron correlation and special relativity, we present the theor. background and the implementation of a new two-component relativistic CI program. It is based on the method of generalized active spaces which has been extended from a nonrelativistic implementation to make use of two-component Hamiltonians and time reversal and double point group symmetry at both the spinor and Slater determinant level. We demonstrate how the great computational effort arising from such a general approach-the treatment of spin-orbit interaction and electron correlation in a fully variational framework-can be markedly reduced by the use of the aforementioned symmetries. Evidence for the performance of the program is given through a no. of calcns. on light systems with a significant spin-orbit splitting in low-lying electronic states and the well-known problem case thallium, which often serves as a rigorous test system in relativistic electronic structure calcns.**40**Saue, T. DIRAC, a relativistic ab initio electronic structure program, 2018. http://www.diracprogram.org.Google ScholarThere is no corresponding record for this reference.**41**Visscher, L.*DIRAC24*, 2024. https://zenodo.org/doi/10.5281/zenodo.10680560.Google ScholarThere is no corresponding record for this reference.**42**Saue, T.; Jensen, H. J. A. Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method.*J. Chem. Phys.*1999,*111*, 6211– 6222, DOI: 10.1063/1.479958Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXmtFKrsbk%253D&md5=54f0466503e395c70c97ab7df3c96ea8Quaternion symmetry in relativistic molecular calculations: the Dirac-Hartree-Fock methodSaue, T.; Jensen, H. J. AaJournal of Chemical Physics (1999), 111 (14), 6211-6222CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A symmetry scheme based on the irreducible co-representations of the full symmetry group of a mol. system is presented for use in relativistic calcns. Consideration of time-reversal symmetry leads to a reformulation of the Dirac-Hartree-Fock equations in terms of quaternion algebra. Further symmetry redns. due to mol. point group symmetry are then manifested by a descent to complex or real algebra. Spatial symmetry is restricted to D2h and subgroups, and it is demonstrated that the Frobenius-Schur test can be used to characterize these groups as a whole. The resulting symmetry scheme automatically provides max. point group and time-reversal symmetry redn. of the computational effort, also when the Fock matrix is constructed in a scalar basis, i.e., from the same type of electron repulsion integrals over symmetry-adapted scalar basis functions as in nonrelativistic theory. An illustrative numerical example is given showing symmetry redns. comparable to the nonrelativistic case.**43**Dyall, K. G.; Fægri, K., Jr.*Introduction to Relativistic Quantum Chemistry*; Oxford University Press, 2007.Google ScholarThere is no corresponding record for this reference.**44**Visscher, L. On the construction of double group molecular symmetry functions.*Chem. Phys. Lett.*1996,*253*, 20– 26, DOI: 10.1016/0009-2614(96)00234-5Google ScholarThere is no corresponding record for this reference.**45**Visscher, L. The Dirac equation in quantum chemistry: Strategies to overcome the current computational problems.*J. Comput. Chem.*2002,*23*, 759– 766, DOI: 10.1002/jcc.10036Google Scholar45https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xjs1Wnsb0%253D&md5=170d5cd0fdc7b2347067c3cad6d4ce6bThe Dirac equation in quantum chemistry: strategies to overcome the current computational problemsVisscher, LucasJournal of Computational Chemistry (2002), 23 (8), 759-766CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A perspective on the use of the relativistic Dirac equation in quantum chem. is given. It is demonstrated that many of the computational problems that plague the current implementations of the different electronic structure methods can be overcome by utilizing the locality of the small component wave function and d. Possible applications of such new and more efficient formulations are discussed.**46**Thyssen, J. Development and Applications of Methods for Correlated Relativistic Calculations of Molecular Properties. Ph.D. thesis, University of Southern Denmark, 2001.Google ScholarThere is no corresponding record for this reference.**47**Li, X.; Paldus, J. Reduced multireference CCSD method: An effective approach to quasidegenerate states.*J. Chem. Phys.*1997,*107*, 6257, DOI: 10.1063/1.474289Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXmsFOhtro%253D&md5=58247434ded7440fd6a789c3f054c993Reduced multireference CCSD method: an effective approach to quasidegenerate statesLi, Xiangzhu; Paldus, JosefJournal of Chemical Physics (1997), 107 (16), 6257-6269CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Std. multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple soln. problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that conc. on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) the construction of a variational CI wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster anal. of this CI wave function providing the information about the higher than pair cluster amplitudes, and (iii) the exploitation of these amplitudes in the so-called externally cor. CCSD procedure. This approach is referred to as the reduced MR (RMR) SS CCSD method and is implemented at the ab initio level and applied to several model systems for which the exact full CI results are available. These include two four electron H4 systems (usually referred to as the H4 and S4 models), an eight electron H8 model, and the singlet-triplet sepn. problem in CH2. It is shown that the RMR CCSD approach produces highly accurate results, is free from intruder state problems, is very general and effective and applicable to both closed and open shell systems.**48**Lyakh, D. I.; Lotrich, V. F.; Bartlett, R. J. The tailored CCSD(T) description of the automerization of cyclobutadiene.*Chem. Phys. Lett.*2011,*501*, 166– 171, DOI: 10.1016/j.cplett.2010.11.058Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXjvVWg&md5=376e9113d921f85860cf34467b3e685eThe 'tailored' CCSD(T) description of the automerization of cyclobutadieneLyakh, Dmitry I.; Lotrich, Victor F.; Bartlett, Rodney J.Chemical Physics Letters (2011), 501 (4-6), 166-171CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)An alternative route to extend the CCSD(T) approach to multireference problems is presented. The well-known defect of the CCSD(T) model in describing the non-dynamic electron correlation effects is remedied by 'tailoring' the underlying coupled-cluster singles and doubles (CCSD) approach and applying the perturbative triples correction to it. The TCCSD(T) approach suggested in the paper has the same computational demands as the CCSD(T) method, though being mostly free from its drawbacks pertinent to multireference (quasidegenerate) situations. To test the approach we calc. the potential energy surface for the automerization of cyclobutadiene where the transition state exhibits a strong multireference character.**49**Melnichuk, A.; Bartlett, R. J. Relaxed active space: Fixing tailored-CC with high order coupled cluster. I.*J. Chem. Phys.*2012,*137*, 214103, DOI: 10.1063/1.4767900Google Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhslOrsrfL&md5=1a4df0dd207e744a42cd87f983feaf18Relaxed active space: Fixing tailored-CC with high order coupled cluster. IMelnichuk, Anna; Bartlett, Rodney J.Journal of Chemical Physics (2012), 137 (21), 214103/1-214103/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Several single ref. (SR-CC) coupled cluster methods are shown to work for traditionally multi-ref. (MR) problems such as bond breaking subject to RHF refs. The correlated methods can successfully describe any MR problem with enough higher order clusters: singles and doubles (CCSD), singles, doubles and triples (CCSDT), singles, doubles, triples, and quadruples (CCSDTQ), etc. However, due to the steep increase in the computational cost, it is not practical to do larger systems or to use large basis sets without active space partitioning. In this study, the orbital space is partitioned into an active space subject to an unambiguous statistical criteria to span the MR behavior which defines an extended space to let the active space relax. The rest is considered the external space. The extended space is treated with CCSDT and the external space with CCSD. An automated scheme for detg. the extended space is presented and evaluated. We build upon the tailored-CC scheme of Hino and address its main shortcoming of neglecting the coupling between the active space and the rest of the orbital space which results in loss of accuracy as well as a pronounced nonparallelism error (NPE). The automated scheme makes it unnecessary for the user to judge whether a chosen active space is sufficient to correctly solve the problem. We illustrate this method for the hydrogen fluoride and fluorine mol. ground state dissocn. potentials using the extended space partitioning methods. Exptl. accuracy for the dissocn. energy is achieved at a small fraction of the cost of doing a full CCSDT calcn. This approach is easily amendable to higher order clusters which are required for double and triple bond breaking and other strongly multi-ref. systems. (c) 2012 American Institute of Physics.**50**Melnichuk, A.; Bartlett, R. J. Relaxed active space: Fixing tailored-CC with high order coupled cluster. II.*J. Chem. Phys.*2014,*140*, 064113, DOI: 10.1063/1.4862676Google Scholar50https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXjvFyms7k%253D&md5=c67b98a2d4ecfaed095330cf6cdc36fbRelaxed active space: Fixing tailored-CC with high order coupled cluster. IIMelnichuk, Ann; Bartlett, Rodney J.Journal of Chemical Physics (2014), 140 (6), 064113/1-064113/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Due to the steep increase in computational cost with the inclusion of higher-connected cluster operators in coupled-cluster applications, it is usually not practical to use such methods for larger systems or basis sets without an active space partitioning. This study generates an active space subject to unambiguous statistical criteria to define a space whose size permits treatment at the CCSDT level. The automated scheme makes it unnecessary for the user to judge whether a chosen active space is sufficient to correctly solve the problem. Two demanding applications are presented: twisted ethylene and the transition states for the bicyclo[1,1,0]butane isomerization. As bi-radicals both systems require at least a CCSDT level of theory for quant. results, for the geometries and energies. (c) 2014 American Institute of Physics.**51**Piecuch, P.; Oliphant, N.; Adamowicz, L. A state-selective multireference coupled-cluster theory employing the single-reference formalism.*J. Chem. Phys.*1993,*99*, 1875– 1900, DOI: 10.1063/1.466179Google Scholar51https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtFWjsrc%253D&md5=85bd8b4c37a2e1763e9032287669e1e5A state-selective multireference coupled-cluster theory employing the single-reference formalismPiecuch, Piotr; Oliphant, Nevin; Adamowicz, LudwikJournal of Chemical Physics (1993), 99 (3), 1875-900CODEN: JCPSA6; ISSN:0021-9606.A new state-selective multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism is described. It is an extension of a simple two-ref. formalism, which the authors presented in the authors' earlier paper [N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991)], and a rigorous formulation of another method of ours, which the authors obtained as an approxn. of the SRCC approach truncated at triple excitations (SRCCSDT) [N. Oliphant and L. Adamowicz, J. Chem. Phys. 96, 3739 (1992)]. The size extensivity of the resulting correlation energies is achieved by employing a SRCC-like ansatz for the multideterminantal wave function. General considerations are supplemented by suggesting a hierarchy of approx. schemes, with the MRCCSD approach (MRCC approach truncated at double excitations from the ref. determinants) representing the most important one. The authors' state-selective MRCCSD theory emerges through a suitable selection of the most essential cluster components appearing in the full SRCCSDTQ method (SRCC method truncated at quadruple excitations), when the latter is applied to quasidegenerate states. The complete set of equations describing the authors' MRCCSD formalism is presented and the possibility of the recursive intermediate factorization [S. A. Kucharski and R. J. Bartlett, Theor. Chim. Acta 80, 387 (1991)] of the authors' approach, leading to an efficient computer algorithm, is discussed.**52**Piecuch, P.; Adamowicz, L. State-selective multireference coupled-cluster theory employing the single-reference formalism: Implementation and application to the H8 model system.*J. Chem. Phys.*1994,*100*, 5792– 5809, DOI: 10.1063/1.467143Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXjtVOksrY%253D&md5=b8df2b4213996bf687d716f91ccaa240State-selective multireference coupled-cluster theory employing the single-reference formalism: implementation and application to the H8 model systemPiecuch, Piotr; Adamowicz, LudwikJournal of Chemical Physics (1994), 100 (8), 5792-809CODEN: JCPSA6; ISSN:0021-9606.The new state-selective (SS) multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism, which was introduced previously (P. Piecuch, et al., 1993), was implemented; results are presented of pilot calcns. for the min. basis-set (MBS) model composed of eight hydrogen atoms in various geometrical arrangements. This model enables a continuous transition between degenerate and nondegenerate regimes. Comparison is made with the results of SR CC calcns. involving double (CCD), single and double (CCSD), single, double, and triple (CCSDT), and single, double, triple, and quadruple (CCSDTQ) excitations. The authors' SS CC energies are also compared with the results of Hilbert space, state-universal (SU) MR CC(S)D calcns., as well as with MR-CI results (with and without Davidson-type corrections), and with exact correlation energies obtained using the full-CI (FCI) method. Along with the ground-state energies, the authors also analyzed the resulting wave functions by examg. some selected cluster components. This anal. enabled the authors to assess the quality of the resulting wave functions. The authors' SS CC theory truncated at double excitations, which emerges through selection of the most essential clusters appearing in the full SR CCSDTQ formalism [SS CCSD (TQ) method], provided equally good results in the nondegenerate and quasidegenerate regions. The difference between the ground-state energy obtained with the SS CCSD(TQ) approach and the FCI energy did not exceed 1.1 milli-hartree over all the geometries considered. This value compares favorably with the max. difference of 2.8 milli-hartrees between the SU CCSD energies and the FCI energies obtained for the same range of geometries. The SS CCSD(T) method, emerging from the SR CCSDT theory through selection of the most essential clusters, was less stable, since it neglected very important semi-internal quadruple excitations. Unlike the genuine multideterminantal SU CC formalism, the authors' SS CC approach was not affected by the intruder-state problem, and its convergence remained satisfactory the in nondegenerate and quasidegenerate regimes.**53**Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy.*Phys. Rev. B*2003,*68*, 195116, DOI: 10.1103/PhysRevB.68.195116Google Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXpvVegs7k%253D&md5=215685d20c465a36d96e9adf4bbb0ea3Optimizing the density-matrix renormalization group method using quantum information entropyLegeza, O.; Solyom, J.Physical Review B: Condensed Matter and Materials Physics (2003), 68 (19), 195116/1-195116/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)In order to optimize the ordering of the lattice sites in the momentum space and quantum chem. versions of the d.-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various mols. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chem. DMRG to a great extent. The effect of lattice site ordering on the no. of block states to be kept during the RG procedure is also investigated.**54**Battaglia, S.; Keller, S.; Knecht, S. Efficient Relativistic Density-Matrix Renormalization Group Implementation in a Matrix-Product Formulation.*J. Chem. Theory Comput.*2018,*14*, 2353– 2369, DOI: 10.1021/acs.jctc.7b01065Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXltFens78%253D&md5=c9dc1a06d66b433bfa2dcbc1a902d153Efficient Relativistic Density-Matrix Renormalization Group Implementation in a Matrix-Product FormulationBattaglia, Stefano; Keller, Sebastian; Knecht, StefanJournal of Chemical Theory and Computation (2018), 14 (5), 2353-2369CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present an implementation of the relativistic quantum-chem. d. matrix renormalization group (DMRG) approach based on a matrix-product formalism. Our approach allows us to optimize matrix product state (MPS) wave functions including a variational description of scalar-relativistic effects and spin-orbit coupling from which we can calc., for example, first-order elec. and magnetic properties in a relativistic framework. While complementing our pilot implementation (Knecht, S. et al., J. Chem. Phys. 2014, 140, 041101), this work exploits all features provided by its underlying nonrelativistic DMRG implementation based on an matrix product state and operator formalism. We illustrate the capabilities of our relativistic DMRG approach by studying the ground-state magnetization, as well as c.d. of a paramagnetic f9 dysprosium complex as a function of the active orbital space employed in the MPS wave function optimization.**55**Szalay, S.; Pfeffer, M.; Murg, V.; Barcza, G.; Verstraete, F.; Schneider, R.; Legeza, Ö. Tensor product methods and entanglement optimization forab initioquantum chemistry.*Int. J. Quantum Chem.*2015,*115*, 1342– 1391, DOI: 10.1002/qua.24898Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXovVags7c%253D&md5=d1f7db2c2c73a2d907e9a290ccb7f22bTensor product methods and entanglement optimization for ab initio quantum chemistrySzalay, Szilard; Pfeffer, Max; Murg, Valentin; Barcza, Gergely; Verstraete, Frank; Schneider, Reinhold; Legeza, OersInternational Journal of Quantum Chemistry (2015), 115 (19), 1342-1391CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The treatment of high-dimensional problems such as the Schroedinger equation can be approached by concepts of tensor product approxn. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods-developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools-can be combined to attack highly challenging problems in quantum chem. The aim of the present paper is to give a pedagogical introduction to the theor. background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum chem." at the Workshop on Theor. Chem., Feb. 18-21, 2014, Mariapfarr, Austria. © 2015 Wiley Periodicals, Inc.**56**Menczer, A.; Legeza, Ö. Massively Parallel Tensor Network State Algorithms on Hybrid CPU-GPU Based Architectures.*arXiv*2023, arXiv:2305.05581v1. DOI: 10.48550/arXiv.2305.05581 .Google ScholarThere is no corresponding record for this reference.**57**Menczer, A.; Legeza, Ö. Boosting the effective performance of massively parallel tensor network state algorithms on hybrid CPU-GPU based architectures via non-Abelian symmetries, 2023. https://arxiv.org/abs/2309.16724.Google ScholarThere is no corresponding record for this reference.**58**Menczer, A.; Kapás, K.; Werner, M. A.; Legeza, O. Two-dimensional quantum lattice models via mode optimized hybrid CPU-GPU density matrix renormalization group method.*Phys. Rev. B*2024,*109*, 195148, DOI: 10.1103/PhysRevB.109.195148Google ScholarThere is no corresponding record for this reference.**59**Menczer, A.; van Damme, M.; Rask, A.; Huntington, L.; Hammond, J.; Xantheas, S. S.; Ganahl, M.; Legeza, Ö. Parallel implementation of the Density Matrix Renormalization Group method achieving a quarter petaFLOPS performance on a single DGX-H100 GPU node.*J. Chem. Theory Comput.*2024,*20*, 8397– 8404, DOI: 10.1021/acs.jctc.4c00903Google ScholarThere is no corresponding record for this reference.**60**Brandejs, J.; Pototschnig, J.; Saue, T. Generating coupled cluster code for modern distributed memory tensor software, 2024. https://arxiv.org/abs/2409.06759.Google ScholarThere is no corresponding record for this reference.**61**Atkinson, B. E.; Hu, H.-S.; Kaltsoyannis, N. Post Hartree–Fock calculations of pnictogen–uranium bonding in EUF3 (E = N–Bi).*Chem. Commun.*2018,*54*, 11100– 11103, DOI: 10.1039/C8CC05581EGoogle ScholarThere is no corresponding record for this reference.**62**TURBOMOLE, v7.52020, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH,1989–2007. https://www.turbomole.org.Google ScholarThere is no corresponding record for this reference.**63**Dyall, K. G. Core correlating basis functions for elements 31–118.*Theor. Chem. Acc.*2012,*131*, 1217, DOI: 10.1007/s00214-012-1217-8Google ScholarThere is no corresponding record for this reference.**64**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 Scholar64https://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.**65**Dunning, T. H., Jr. ANL vibration–rotation analysis program for diatomic molecules, 1979.Google ScholarThere is no corresponding record for this reference.**66**Visscher, L. Approximate molecular Dirac-Coulomb calculations using a simple Coulombic correction.*Theor. Chem. Acc.*1997,*98*, 68, DOI: 10.1007/s002140050280Google Scholar66https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXotFamu74%253D&md5=ac65458dcfe7cc47f433e92757371b92Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correctionVisscher, LucasTheoretical Chemistry Accounts (1997), 98 (2-3), 68-70CODEN: TCACFW; ISSN:1432-881X. (Springer-Verlag)A simple point-charge model is used to correct mol. 4-component Dirac-Coulomb calcns. which neglect 2-electron integrals over the small components of the wave function. The calcd. valence properties show no degeneration relative to the full calcn., while a speed-up factor of 3 is obtained.**67**Andrews, L.; Wang, X.; Lindh, R.; Roos, B.; Marsden, C. Simple NUF3 and PUF3Molecules with Triple Bonds to Uranium.*Angew. Chem., Int. Ed.*2008,*47*, 5366– 5370, DOI: 10.1002/anie.200801120Google ScholarThere is no corresponding record for this reference.**68**Van Gundy, R. A. Electronic Structure of Metal-Containing Diatomic Ions. Ph.D. thesis, Faculty of the James T. Laney School of Graduate Studies of Emory University, 2018.Google ScholarThere is no corresponding record for this reference.**69**King, D. M.; Liddle, S. T. Progress in molecular uranium-nitride chemistry.*Coord. Chem. Rev.*2014,*266–267*, 2– 15, DOI: 10.1016/j.ccr.2013.06.013Google Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXjsVChtrc%253D&md5=ddd0c7bd4cb450c7582ebd60bea3fe05Progress in molecular uranium-nitride chemistryKing, David M.; Liddle, Stephen T.Coordination Chemistry Reviews (2014), 266-267 (), 2-15CODEN: CCHRAM; ISSN:0010-8545. (Elsevier B.V.)A review. The coordination, organometallic, and materials chem. of uranium nitride has long been an important facet of actinide chem. Following matrix isolation expts. and computational characterization, mol., soln.-based uranium chem. has developed significantly in the last decade or so culminating most recently in the isolation of the first examples of long-sought terminal uranium nitride linkages. Herein, the field is reviewed with an emphasis on well-defined mol. species.**70**Balasubramanian, S. G. TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations.*J. Chem. Phys.*2020,*152*, 184107, DOI: 10.1063/5.0004635Google ScholarThere is no corresponding record for this reference.**71**Furness, J. W.; Kaplan, A. D.; Ning, J.; Perdew, J. P.; Sun, J. Accurate and Numerically Efficient r2SCAN Meta-Generalized Gradient Approximation.*J. Phys. Chem. Lett.*2020,*11*, 8208– 8215, DOI: 10.1021/acs.jpclett.0c02405Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhslequ77N&md5=49adb31d49e1e53910d87275f6400ae9Accurate and Numerically Efficient r2SCAN Meta-Generalized Gradient ApproximationFurness, James W.; Kaplan, Aaron D.; Ning, Jinliang; Perdew, John P.; Sun, JianweiJournal of Physical Chemistry Letters (2020), 11 (19), 8208-8215CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)The recently proposed rSCAN functional [J. Chem. Phys., 2019, 150, 161101] is a regularized form of the SCAN functional [Phys. Rev. Lett., 2015, 115, 036402] that improves SCAN's numerical performance at the expense of breaking constraints known from the exact exchange-correlation functional. We construct a new meta-generalized gradient approxn. by restoring exact constraint adherence to rSCAN. The resulting functional maintains rSCAN's numerical performance while restoring the transferable accuracy of SCAN.**72**Holzer, C.; Franzke, Y. J.; Kehry, M. Assessing the Accuracy of Local Hybrid Density Functional Approximations for Molecular Response Properties.*J. Chem. Theory Comput.*2021,*17*, 2928– 2947, DOI: 10.1021/acs.jctc.1c00203Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXpslejsrY%253D&md5=9a7b4bbcbd5a5b3dd35fce208f203513Assessing the Accuracy of Local Hybrid Density Functional Approximations for Molecular Response PropertiesHolzer, Christof; Franzke, Yannick J.; Kehry, MaxJournal of Chemical Theory and Computation (2021), 17 (5), 2928-2947CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A comprehensive overview of the performance of local hybrid functionals for mol. properties like excited states, ionization potentials within the GW framework, polarizabilities, magnetizabilities, NMR chem. shifts, and NMR spin-spin coupling consts. is presented. We apply the generalization of the kinetic energy, τ, with the paramagnetic c.d. to all magnetic properties and the excitation energies from time-dependent d. functional theory. This restores gauge invariance for these properties. Different ansatze for local mixing functions such as the iso-orbital indicator, the correlation length, the Gorling-Levy second-order limit, and the spin polarization are compared. For the latter, we propose a modified version of the corresponding hyper-generalized gradient approxn. functional of Perdew, Staroverov, Tao, and Scuseria (PSTS) to allow for a numerically stable evaluation of the exchange-correlation kernel and hyperkernel. The PSTS functional leads to a very consistent improvement compared to the related TPSSh functional. It is further shown that the "best" choice of the local mixing function depends on the studied property and mol. class. While functionals based on the iso-orbital indicator lead to rather accurate excitation energies and ionization energies, the results are less impressive for NMR properties, for which a considerable dependence on the considered mol. test set and nuclei is obsd. Johnson's local hybrid functional based on the correlation length yields remarkable results for NMR shifts of compds. featuring heavy elements and also for the excitation energies of org. compds.**73**Perdew, J. P.; Staroverov, V. N.; Tao, J.; Scuseria, G. E. Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfaction.*Phys. Rev. A*2008,*78*, 052513, DOI: 10.1103/PhysRevA.78.052513Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhsVGqtbbP&md5=4b1748d8e8d9deb01f650564d60ec7e5Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfactionPerdew, John P.; Staroverov, Viktor N.; Tao, Jianmin; Scuseria, Gustavo E.Physical Review A: Atomic, Molecular, and Optical Physics (2008), 78 (5, Pt. A), 052513/1-052513/13CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We construct a nonlocal d. functional approxn. with full exact exchange, while preserving the constraint-satisfaction approach and justified error cancellations of simpler semilocal functionals. This is achieved by interpolating between different approxns. suitable for two extreme regions of the electron d. In a "normal" region, the exact exchange-correlation hole d. around an electron is semilocal because its spatial range is reduced by correlation and because it integrates over a narrow range to -1. These regions are well described by popular semilocal approxns. (many of which have been constructed nonempirically), because of proper accuracy for a slowly varying d. or because of error cancellation between exchange and correlation. "Abnormal" regions, where nonlocality is unveiled, include those in which exchange can dominate correlation (one-electron, nonuniform high d., and rapidly varying limits), and those open subsystems of fluctuating electron no. over which the exact exchange-correlation hole integrates to a value greater than -1. Regions between these extremes are described by a hybrid functional mixing exact and semilocal exchange energy densities locally, i.e., with a mixing fraction that is a function of position r and a functional of the d. Because our mixing fraction tends to 1 in the high-d. limit, we employ full exact exchange according to the rigorous definition of the exchange component of any exchange-correlation energy functional. Use of full exact exchange permits the satisfaction of many exact constraints, but the nonlocality of exchange also requires balanced nonlocality of correlation. We find that this nonlocality can demand at least five empirical parameters, corresponding roughly to the four kinds of abnormal regions. Our local hybrid functional is perhaps the first accurate fourth-rung d. functional or hyper-generalized gradient approxn., with full exact exchange, that is size-consistent in the way that simpler functionals are. It satisfies other known exact constraints, including exactness for all one-electron densities, and provides an excellent fit to the 223 mol. enthalpies of formation of the G3/99 set and the 42 reaction barrier heights of the BH42/03 set, improving both (but esp. the latter) over most semilocal functionals and global hybrids. Exact constraints, phys. insights, and paradigm examples hopefully suppress "overfitting.".

## Cited By

This article has not yet been cited by other publications.

## Article Views

## Altmetric

## Citations

Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.

Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.

The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated.

## Recommended Articles

## References

This article references 73 other publications.

**1**White, S. R. Density matrix formulation for quantum renormalization groups.*Phys. Rev. Lett.*1992,*69*, 2863– 2866, DOI: 10.1103/PhysRevLett.69.28631https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2sfptF2isg%253D%253D&md5=51e8562b250f575cd902524cde61c5d1Density matrix formulation for quantum renormalization groupsWhitePhysical review letters (1992), 69 (19), 2863-2866 ISSN:.There is no expanded citation for this reference.**2**White, S. R.; Martin, R. L. Ab initio quantum chemistry using the density matrix renormalization group.*J. Chem. Phys.*1999,*110*, 4127– 4130, DOI: 10.1063/1.4782952https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXhtF2gtbw%253D&md5=ae0c47542a0ddc08171b93f29693e51fAb initio quantum chemistry using the density matrix renormalization groupWhite, Steven R.; Martin, Richard L.Journal of Chemical Physics (1999), 110 (9), 4127-4130CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper we describe how the d. matrix renormalization group can be used for quantum chem. calcns. for mols., as an alternative to traditional methods, such as CI or coupled cluster approaches. As a demonstration of the potential of this approach, we present results for the H2O mol. in a std. Gaussian basis. Results for the total energy of the system compare favorably with the best traditional quantum chem. methods.**3**Chan, G. K.-L.; Head-Gordon, M. Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group.*J. Chem. Phys.*2002,*116*, 4462– 4476, DOI: 10.1063/1.14494593https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhslCjsrs%253D&md5=f1249212b5c6414a901ff5c8a4a64542Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization groupChan, Garnet Kin-Lic; Head-Gordon, MartinJournal of Chemical Physics (2002), 116 (11), 4462-4476CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the recently developed D. Matrix Renormalization Group (DMRG) algorithm in the context of quantum chem. In contrast to traditional approaches, this algorithm is believed to yield arbitrarily high accuracy in the energy with only polynomial computational effort. We describe in some detail how this is achieved. We begin by introducing the principles of the renormalization procedure, and how one formulates an algorithm for use in quantum chem. The renormalization group algorithm is then interpreted in terms of familiar quantum chem. concepts, and its numerical behavior, including its convergence and computational cost, are studied using both model and real systems. The asymptotic convergence of the algorithm is derived. Finally, we examine the performance of the DMRG on widely studied chem. problems, such as the water mol., the twisting barrier of ethene, and the dissocn. of nitrogen. In all cases, the results are favorably comparable with the best existing quantum chem. methods, and particularly so when the nondynamical correlation is strong. Some perspectives for future development are given.**4**Legeza, Ö.; Röder, J.; Hess, B. A. Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach.*Phys. Rev. B*2003,*67*, 125114, DOI: 10.1103/PhysRevB.67.1251144https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXivFaqsbY%253D&md5=0f5650c8557f55974dea14ef3e8a41d3Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approachLegeza, O.; Roder, J.; Hess, B. A.Physical Review B: Condensed Matter and Materials Physics (2003), 67 (12), 125114/1-125114/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)We have applied the momentum space version of the d.-matrix renormalization-group method (k-DMRG) in quantum chem. in order to study the accuracy of the algorithm in this new context. We have shown numerically that it is possible to det. the desired accuracy of the method in advance of the calcns. by dynamically controlling the truncation error and the no. of block states using a novel protocol that we dubbed dynamical block state selection protocol. The relationship between the real error and truncation error has been studied as a function of the no. of orbitals and the fraction of filled orbitals. We have calcd. the ground state of the mols. CH2, H2O, and F2 as well as the first excited state of CH2. Our largest calcns. were carried out with 57 orbitals, the largest no. of block states was 1500-2000, and the largest dimensions of the Hilbert space of the superblock configuration was 800 000-1 200 000.**5**Schollwöck, U. The density-matrix renormalization group in the age of matrix product states.*Annals of Physics*2011,*326*, 96– 192, DOI: 10.1016/j.aop.2010.09.012There is no corresponding record for this reference.**6**Olivares-Amaya, R.; Hu, W.; Nakatani, N.; Sharma, S.; Yang, J.; Chan, G. K.-L. The ab-initio density matrix renormalization group in practice.*J. Chem. Phys.*2015,*142*, 034102, DOI: 10.1063/1.49053296https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXovFSmsQ%253D%253D&md5=4e30986e7c45a42b1df2a78031f17c58The ab-initio density matrix renormalization group in practiceOlivares-Amaya, Roberto; Hu, Weifeng; Nakatani, Naoki; Sharma, Sandeep; Yang, Jun; Chan, Garnet Kin-LicJournal of Chemical Physics (2015), 142 (3), 034102/1-034102/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The ab-initio d. matrix renormalization group (DMRG) is a tool that can be applied to a wide variety of interesting problems in quantum chem. Here, we examine the d. matrix renormalization group from the vantage point of the quantum chem. user. What kinds of problems is the DMRG well-suited to. What are the largest systems that can be treated at practical cost. What sort of accuracies can be obtained, and how do we reason about the computational difficulty in different mols.. By examg. a diverse benchmark set of mols.: π-electron systems, benchmark main-group and transition metal dimers, and the Mn-oxo-salen and Fe-porphine organometallic compds., we provide some answers to these questions, and show how the d. matrix renormalization group is used in practice. (c) 2015 American Institute of Physics.**7**Kurashige, Y.; Yanai, T. Second-order perturbation theory with a density matrix renormalization group self-consistent field reference function: Theory and application to the study of chromium dimer.*J. Chem. Phys.*2011,*135*, 094104, DOI: 10.1063/1.36294547https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtFajtr%252FF&md5=3555ac1743fc963a44b9c86c08168517Second-order perturbation theory with a density matrix renormalization group self-consistent field reference function: Theory and application to the study of chromium dimerKurashige, Yuki; Yanai, TakeshiJournal of Chemical Physics (2011), 135 (9), 094104/1-094104/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a second-order perturbation theory based on a d. matrix renormalization group SCF (DMRG-SCF) ref. function. The method reproduces the soln. of the complete active space with second-order perturbation theory (CASPT2) when the DMRG ref. function is represented by a sufficiently large no. of renormalized many-body basis, thereby being named DMRG-CASPT2 method. The DMRG-SCF is able to describe non-dynamical correlation with large active space that is insurmountable to the conventional CASSCF method, while the second-order perturbation theory provides an efficient description of dynamical correlation effects. The capability of our implementation is demonstrated for an application to the potential energy curve of the chromium dimer, which is one of the most demanding multireference systems that require best electronic structure treatment for non-dynamical and dynamical correlation as well as large basis sets. The DMRG-CASPT2/cc-pwCV5Z calcns. were performed with a large (3d double-shell) active space consisting of 28 orbitals. Our approach using large-size DMRG ref. addressed the problems of why the dissocn. energy is largely overestimated by CASPT2 with the small active space consisting of 12 orbitals (3d4s), and also is oversensitive to the choice of the zeroth-order Hamiltonian. (c) 2011 American Institute of Physics.**8**Saitow, M.; Kurashige, Y.; Yanai, T. Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave function.*J. Chem. Phys.*2013,*139*, 044118, DOI: 10.1063/1.48166278https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhtFygsbnL&md5=a30f97f373ba1a069bbce1c23a48e178Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave functionSaitow, Masaaki; Kurashige, Yuki; Yanai, TakeshiJournal of Chemical Physics (2013), 139 (4), 044118/1-044118/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We report development of the multireference CI (MRCI) method that can use active space scalable to much larger size refs. than has previously been possible. The recent development of the d. matrix renormalization group (DMRG) method in multireference quantum chem. offers the ability to describe static correlation in a large active space. The present MRCI method provides a crit. correction to the DMRG ref. by including high-level dynamic correlation through the CI treatment. When the DMRG and MRCI theories are combined (DMRG-MRCI), the full internal contraction of the ref. in the MRCI ansatz, including contraction of semi-internal states, plays a central role. However, it is thought to involve formidable complexity because of the presence of the five-particle rank reduced-d. matrix (RDM) in the Hamiltonian matrix elements. To address this complexity, we express the Hamiltonian matrix using commutators, which allows the five-particle rank RDM to be canceled out without any approxn. Then we introduce an approxn. to the four-particle rank RDM by using a cumulant reconstruction from lower-particle rank RDMs. A computer-aided approach is employed to derive the exceedingly complex equations of the MRCI in tensor-contracted form and to implement them into an efficient parallel computer code. This approach extends to the size-consistency-cor. variants of MRCI, such as the MRCI+Q, MR-ACPF, and MR-AQCC methods. We demonstrate the capability of the DMRG-MRCI method in several benchmark applications, including the evaluation of single-triplet gap of free-base porphyrin using 24 active orbitals. (c) 2013 American Institute of Physics.**9**Wouters, S.; Nakatani, N.; Van Neck, D.; Chan, G. K.-L. Thouless theorem for matrix product states and subsequent post density matrix renormalization group methods.*Phys. Rev. B*2013,*88*, 075122, DOI: 10.1103/PhysRevB.88.0751229https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhsFykt7fN&md5=d0c396e966684bcc99590f31a9d885e6Thouless theorem for matrix product states and subsequent post density matrix renormalization group methodsWouters, Sebastian; Nakatani, Naoki; Van Neck, Dimitri; Chan, Garnet Kin-LicPhysical Review B: Condensed Matter and Materials Physics (2013), 88 (7), 075122/1-075122/14CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)The similarities between Hartree-Fock (HF) theory and the d. matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function Ansatz. Linearization of the time-dependent variational principle near a variational min. allows to derive the RPA (RPA). We show that the nonredundant parameterization of the matrix product state (MPS) tangent space leads to the Thouless theorem for MPS, i.e., an explicit nonredundant parameterization of the entire MPS manifold, starting from a specific MPS ref. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approxn. (TDA), the CI (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS Ansatz with single and double excitations to improve on the ground state and to calc. low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calcns. of the RPA-MPS correlation energy. With the long-range quantum chem. PPP Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.**10**Yanai, T.; Chan, G. K.-L. Canonical transformation theory for multireference problems.*J. Chem. Phys.*2006,*124*, 194106, DOI: 10.1063/1.219641010https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XltVOjsbs%253D&md5=15d27491b9be26ab2fe5888182f60b48Canonical transformation theory for multireference problemsYanai, Takeshi; Chan, Garnet Kin-LicJournal of Chemical Physics (2006), 124 (19), 194106/1-194106/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We propose a theory to describe dynamic correlations in bonding situations where there is also significant nondynamic character. We call this the canonical transformation (CT) theory. When combined with a suitable description of nondynamic correlation, such as given by a complete-active-space SCF (CASSCF) or d. matrix renormalization group wave function, it provides a theory to describe bonding situations across the entire potential energy surface with quant. accuracy for both dynamic and nondynamic correlation. The canonical transformation theory uses a unitary exponential ansatz, is size consistent, and has a computational cost of the same order as a single-ref. coupled cluster theory with the same level of excitations. Calcns. using the CASSCF based CT method with single and double operators for the potential energy curves for water and nitrogen mols., the BeH2 insertion reaction, and hydrogen fluoride and boron hydride bond breaking, consistently yield quant. accuracies typical of equil. region coupled cluster theory, but across all geometries, and better than obtained with multireference perturbation theory.**11**Ren, J.; Yi, Y.; Shuai, Z. Inner Space Perturbation Theory in Matrix Product States: Replacing Expensive Iterative Diagonalization.*J. Chem. Theory Comput.*2016,*12*, 4871– 4878, DOI: 10.1021/acs.jctc.6b0069611https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsVKrt7nM&md5=999a0e303af01c951094be2ef7b64ee6Inner Space Perturbation Theory in Matrix Product States: Replacing Expensive Iterative DiagonalizationRen, Jiajun; Yi, Yuanping; Shuai, ZhigangJournal of Chemical Theory and Computation (2016), 12 (10), 4871-4878CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We propose an inner space perturbation theory (isPT) to replace the expensive iterative diagonalization in the std. d. matrix renormalization group theory (DMRG). The retained reduced d. matrix eigenstates are partitioned into the active and secondary space. The first-order wave function and the second- and third-order energies are easily computed by using one step Davidson iteration. Our formulation has several advantages including (i) keeping a balance between the efficiency and accuracy, (ii) capturing more entanglement with the same amt. of computational time, (iii) recovery of the std. DMRG when all the basis states belong to the active space. Numerical examples for the polyacenes and periacene show that the efficiency gain is considerable and the accuracy loss due to the perturbation treatment is very small, when half of the total basis states belong to the active space. Moreover, the perturbation calcns. converge in all our numerical examples.**12**Beran, P.; Matoušek, M.; Hapka, M.; Pernal, K.; Veis, L. Density matrix renormalization group with dynamical correlation via adiabatic connection.*J. Chem. Theory Comput.*2021,*17*, 7575, DOI: 10.1021/acs.jctc.1c0089612https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXisVGrsLjL&md5=6dfdecdd844b5a539f1bd28e0d77e612Density Matrix Renormalization Group with Dynamical Correlation via Adiabatic ConnectionBeran, Pavel; Matousek, Mikulas; Hapka, Michal; Pernal, Katarzyna; Veis, LiborJournal of Chemical Theory and Computation (2021), 17 (12), 7575-7585CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The quantum chem. version of the d. matrix renormalization group (DMRG) method has established itself as one of the methods of choice for calcns. of strongly correlated mol. systems. Despite its great ability to capture strong electronic correlation in large active spaces, it is not suitable for computations of dynamical electron correlation. In this work, we present a new approach to the electronic structure problem of strongly correlated mols., in which DMRG is responsible for a proper description of the strong correlation, whereas dynamical correlation is computed via the recently developed adiabatic connection (AC) technique which requires only up to two-body active space reduced d. matrixes. We report the encouraging results of this approach on typical candidates for DMRG computations, namely, n-acenes (n = 2 → 7), Fe(II)-porphyrin, and the Fe3S4 cluster.**13**Barcza, G.; Werner, M. A.; Zaránd, G.; Pershin, A.; Benedek, Z.; Legeza, Ö.; Szilvási, T. Toward Large-Scale Restricted Active Space Calculations Inspired by the Schmidt Decomposition.*J. Phys. Chem. A*2022,*126*, 9709– 9718, DOI: 10.1021/acs.jpca.2c0595213https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XjtFSht7nO&md5=5f4a7788606b693ab9dca3e86f4e1314Toward large-scale restricted active space calculations inspired by the Schmidt decompositionBarcza, Gergely; Werner, Miklos Antal; Zarand, Gergely; Pershin, Anton; Benedek, Zsolt; Legeza, Ors; Szilvasi, TiborJournal of Physical Chemistry A (2022), 126 (51), 9709-9718CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present an alternative, memory-efficient, Schmidt decompn.-based description of the inherently bipartite restricted active space (RAS) scheme, which can be implemented effortlessly within the d. matrix renormalization group (DMRG) method via the dynamically extended active space procedure. Benchmark calcns. are compared against state-of-the-art results of C2 and Cr2, which are notorious for their multireference character. Our results for ground and excited states together with spectroscopic consts. demonstrate that the proposed novel approach, dubbed as DMRG-RAS, which is variational and free of uncontrolled method errors, has the potential to outperfom conventional methods for strongly correlated mols.**14**Friesecke, G.; Barcza, G.; Legeza, O. Predicting the FCI energy of large systems to chemical accuracy from restricted active space density matrix renormalization group calculations.*J. Chem. Theory Comput.*2024,*20*, 87– 102, DOI: 10.1021/acs.jctc.3c01001There is no corresponding record for this reference.**15**Veis, L.; Antalík, A.; Brabec, J.; Neese, F.; Legeza, Ö.; Pittner, J. Coupled Cluster Method with Single and Double Excitations Tailored by Matrix Product State Wave Functions.*J. Phys. Chem. Lett.*2016,*7*, 4072– 4078, DOI: 10.1021/acs.jpclett.6b0190815https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsFOktrbL&md5=6f0a8d79fff88c257aef0327abd74296Coupled Cluster Method with Single and Double Excitations Tailored by Matrix Product State Wave FunctionsVeis, Libor; Antalik, Andrej; Brabec, Jiri; Neese, Frank; Legeza, Ors; Pittner, JiriJournal of Physical Chemistry Letters (2016), 7 (20), 4072-4078CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)In the past decade, the quantum chem. version of the d. matrix renormalization group (DMRG) method has established itself as the method of choice for calcns. of strongly correlated mol. systems. Despite its favorable scaling, it is in practice not suitable for computations of dynamic correlation. We present a novel method for accurate "post-DMRG" treatment of dynamic correlation based on the tailored coupled cluster (CC) theory in which the DMRG method is responsible for the proper description of nondynamic correlation, whereas dynamic correlation is incorporated through the framework of the CC theory. We illustrate the potential of this method on prominent multireference systems, in particular, N2 and Cr2 mols. and also oxo-Mn(Salen), for which we have performed the first post-DMRG computations in order to shed light on the energy ordering of the lowest spin states.**16**Kinoshita, T.; Hino, O.; Bartlett, R. J. Coupled-cluster method tailored by configuration interaction.*J. Chem. Phys.*2005,*123*, 074106, DOI: 10.1063/1.200025116https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpsleht7w%253D&md5=7c005bedda722c576125ea4cec764785Coupled-cluster method tailored by configuration interactionKinoshita, Tomoko; Hino, Osamu; Bartlett, Rodney J.Journal of Chemical Physics (2005), 123 (7), 074106/1-074106/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A method is presented which combines coupled cluster (CC) and CI to describe accurately potential-energy surfaces (PESs). We use the cluster amplitudes extd. from the complete active space CI calcn. to manipulate nondynamic correlation to tailor a single ref. CC theory (TCC). The dynamic correlation is then incorporated through the framework of the CC method. We illustrate the method by describing the PESs for HF, H2O, and N2 mols. which involve single, double, and triple bond-breaking processes. To the dissocn. limit, this approach yields far more accurate PESs than those obtained from the conventional CC method and the addnl. computational cost is negligible compared with the CC calcn. steps. We anticipate that TCC offers an effective and generally applicable approach for many problems.**17**Hino, O.; Kinoshita, T.; Chan, G. K.-L.; Bartlett, R. J. Tailored coupled cluster singles and doubles method applied to calculations on molecular structure and harmonic vibrational frequencies of ozone.*J. Chem. Phys.*2006,*124*, 114311, DOI: 10.1063/1.218077517https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XivV2jtrs%253D&md5=57806470896d5932ad362d871712636dTailored coupled cluster singles and doubles method applied to calculations on molecular structure and harmonic vibrational frequencies of ozoneHino, Osamu; Kinoshita, Tomoko; Chan, Garnet Kin-Lic; Bartlett, Rodney J.Journal of Chemical Physics (2006), 124 (11), 114311/1-114311/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)To assess the sepn. of dynamic and nondynamic correlations and orbital choice, we calc. the mol. structure and harmonic vibrational frequencies of ozone with the recently developed tailored coupled cluster singles and doubles method (TCCSD). We employ the Hartree-Fock and complete active space (CAS) SCF orbitals to perform TCCSD calcns. When using the Hartree-Fock orbitals, it is difficult to reproduce the exptl. vibrational frequency of the asym. stretching mode. On the other hand, the TCCSD based on the CASSCF orbitals in a correlation consistent polarized valence triple zeta basis yields excellent results with the two sym. vibrations differing from the exptl. harmonic values by 2 cm-1 and the asym. vibration differing by 9 cm-1.**18**Veis, L.; Antalík, A.; Brabec, J.; Neese, F.; Legeza, Ö.; Pittner, J.*J. Phys. Chem. Lett.*2017,*8*, 291, DOI: 10.1021/acs.jpclett.6b02912There is no corresponding record for this reference.**19**Faulstich, F. M.; Laestadius, A.; Legeza, Ö.; Schneider, R.; Kvaal, S. Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry.*SIAM Journal on Numerical Analysis*2019,*57*, 2579– 2607, DOI: 10.1137/18M1171436There is no corresponding record for this reference.**20**Faulstich, F. M.; Máté, M.; Laestadius, A.; Csirik, M. A.; Veis, L.; Antalik, A.; Brabec, J.; Schneider, R.; Pittner, J.; Kvaal, S.; Legeza, Ö. Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer.*J. Chem. Theory Comput.*2019,*15*, 2206– 2220, DOI: 10.1021/acs.jctc.8b0096020https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXjsleju7c%253D&md5=a1b71dd62855082fb746af6f6fbcb493Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen DimerFaulstich, Fabian M.; Mate, Mihaly; Laestadius, Andre; Csirik, Mihaly Andras; Veis, Libor; Antalik, Andrej; Brabec, Jiri; Schneider, Reinhold; Pittner, Jiri; Kvaal, Simen; Legeza, OrsJournal of Chemical Theory and Computation (2019), 15 (4), 2206-2220CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In this article, we investigate the numerical and theor. aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing math. anal. is here elaborated in a quantum chem. framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO-LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the d. matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the d. matrix renormalization group-tailored coupled-cluster error.**21**Leszczyk, A.; Máté, M.; Legeza, Ö.; Boguslawski, K. Assessing the accuracy of tailored coupled cluster methods corrected by electronic wave functions of polynomial cost.*J. Chem. Theory Comp.*2022,*18*, 96, DOI: 10.1021/acs.jctc.1c00284There is no corresponding record for this reference.**22**Neese, F. The ORCA program system.*WIREs Computational Molecular Science*2012,*2*, 73– 78, DOI: 10.1002/wcms.8122https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvFGls7s%253D&md5=a753e33a6f9a326553295596f5c754e5The ORCA program systemNeese, FrankWiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (1), 73-78CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. ORCA is a general-purpose quantum chem. program package that features virtually all modern electronic structure methods (d. functional theory, many-body perturbation and coupled cluster theories, and multireference and semiempirical methods). It is designed with the aim of generality, extendibility, efficiency, and user friendliness. Its main field of application is larger mols., transition metal complexes, and their spectroscopic properties. ORCA uses std. Gaussian basis functions and is fully parallelized. The article provides an overview of its current possibilities and documents its efficiency.**23**Antalik, A.; Veis, L.; Brabec, J.; Demel, O.; Legeza, O.; Pittner, J. Toward the efficient local tailored coupled cluster approximation and the peculiar case of oxo-Mn(Salen).*J. Chem. Phys.*2019,*151*, 084112, DOI: 10.1063/1.5110477There is no corresponding record for this reference.**24**Lang, J.; Antalik, A.; Veis, L.; Brandejs, J.; Brabec, J.; Legeza, O.; Pittner, J. Near-linear Scaling in DMRG-based Tailored Coupled Clusters: An Implementation of DLPNO-TCCSD and DLPNO-TCCSD(T).*J. Chem. Theor. Comput.*2020,*16*, 3028, DOI: 10.1021/acs.jctc.0c00065There is no corresponding record for this reference.**25**Antalik, A.; Nachtigallova, D.; Lo, R.; Matousek, M.; LAng, J.; Legeza, O.; Pittner, J.; Hobza, P.; Veis, L. Ground state of the Fe(II)-porphyrin model system corresponds to the quintet state: DFT, DMRG-TCCSD and DMRG-TCCSD(T) computations.*Phys. Chem. Chem. Phys.*2020,*22*, 17033, DOI: 10.1039/D0CP03086DThere is no corresponding record for this reference.**26**Brandejs, J.; Višňák, J.; Veis, L.; Maté, M.; Legeza, O.; Pittner, J. Toward DMRG-tailored coupled cluster method in the 4c-relativistic domain.*J. Chem. Phys.*2020,*152*, 174107, DOI: 10.1063/1.514497426https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXovFGktr8%253D&md5=0d571efd9ea26884e1055dcd374a4918Toward DMRG-tailored coupled cluster method in the 4c-relativistic domainBrandejs, Jan; Visnak, Jakub; Veis, Libor; Mate, Mihaly; Legeza, Ors; Pittner, JiriJournal of Chemical Physics (2020), 152 (17), 174107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)There are three essential problems in computational relativistic chem.: Electrons moving at relativistic speeds, close lying states, and dynamical correlation. Currently available quantum-chem. methods are capable of solving systems with one or two of these issues. However, there is a significant class of mols. in which all the three effects are present. These are the heavier transition metal compds., lanthanides, and actinides with open d or f shells. For such systems, sufficiently accurate numerical methods are not available, which hinders the application of theor. chem. in this field. In this paper, we combine two numerical methods in order to address this challenging class of mols. These are the relativistic versions of coupled cluster methods and the d. matrix renormalization group (DMRG) method. To the best of our knowledge, this is the first relativistic implementation of the coupled cluster method externally cor. by DMRG. The method brings a significant redn. of computational costs as we demonstrate on the system of TlH, AsH, and SbH. (c) 2020 American Institute of Physics.**27**Wormit, M.; Olejniczak, M.łg.; Deppenmeier, A.-L.; Borschevsky, A.; Saue, T.; Schwerdtfeger, P. Strong enhancement of parity violation effects in chiral uranium compounds.*Phys. Chem. Chem. Phys.*2014,*16*, 17043– 17051, DOI: 10.1039/C4CP01904K27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVKgsr7L&md5=dddf4e351522977f2dd9a619151d4a44Strong enhancement of parity violation effects in chiral uranium compoundsWormit, Michael; Olejniczak, Malgorzata; Deppenmeier, Anna-Lena; Borschevsky, Anastasia; Saue, Trond; Schwerdtfeger, PeterPhysical Chemistry Chemical Physics (2014), 16 (32), 17043-17051CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)The effects of parity violation (PV) on the vibrational transitions of chiral uranium compds. of the type N≡UXYZ and N≡UHXY (X, Y, Z = F, Cl, Br, I) are analyzed by means of exact two-component relativistic (X2C) Hartree-Fock and d. functional calcns. using NUFClI and NUHFI as representative examples. The PV contributions to the vibrational transitions were found to be in the Hz range, larger than for any of the earlier proposed chiral mols. Thus, these systems are very promising candidates for future exptl. PV measurements. A detailed comparison of the N≡UHFI and the N≡WHFI homologues reveals that subtle electronic structure effects, rather than exclusively a simple Z5 scaling law, are the cause of the strong enhancement in PV contributions of the chiral uranium mols.**28**Jiang, W.; Wilson, A. K. Multireference composite approaches for the accurate study of ground and excited electronic states: C2, N2, and O2.*J. Chem. Phys.*2011,*134*, 034101, DOI: 10.1063/1.351403128https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmvFWrtQ%253D%253D&md5=d1460e6260a528382a9fbfd151f2d1d5Multireference composite approaches for the accurate study of ground and excited electronic states: C2, N2, and O2Jiang, Wanyi; Wilson, Angela K.Journal of Chemical Physics (2011), 134 (3), 034101/1-034101/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A multiref. analog of the correlation consistent composite approach (MR-ccCA) based on complete active space with 2nd-order perturbation theory (CASPT2) was used in a study of the ground and valence excited states of C2, N2, and O2. The performance of different 2nd-order multiref. perturbation theory methods including 2nd-order n-electron valence state perturbation theory, 2nd-order multiref. Moller-Plesset, and 2nd-order generalized van Vleck perturbation theory was analyzed as potential alternatives to CASPT2 within MR-ccCA. The MR-ccCA-P predicts spectroscopic consts. with overall mean abs. deviations from exptl. values of 0.0006 Å, 7.0 cm-1, and 143 cm-1 for equil. bond length (re), harmonic frequency (ωe), and term values (Te), resp., which are comparable to the predictions by more computationally costly multiref. CI-based methods. (c) 2011 American Institute of Physics.**29**Mintz, B.; Williams, T. G.; Howard, L.; Wilson, A. K. Computation of potential energy surfaces with the multireference correlation consistent composite approach.*J. Chem. Phys.*2009,*130*, 234104, DOI: 10.1063/1.314938729https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXnsFClurw%253D&md5=e336a765219902a7b85523063806534cComputation of potential energy surfaces with the multireference correlation consistent composite approachMintz, Benjamin; Williams, T. Gavin; Howard, Levi; Wilson, Angela K.Journal of Chemical Physics (2009), 130 (23), 234104/1-234104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A multireference composite method that is based on the correlation consistent Composite Approach (ccCA) is introduced. The developed approach, multireference ccCA, was utilized to compute the potential energy surfaces (PESs) of N2 and C2, which provide rigorous tests for multireference composite methods due to the large multireference character that must be correctly described as the mols. dissoc. As well, PESs provide a stringent test of a composite method because all components of the method must work in harmony for an appropriate, smooth representation across the entire surface. (c) 2009 American Institute of Physics.**30**Chan, G. K.-L.; Kállay, M.; Gauss, J. State-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curve.*J. Chem. Phys.*2004,*121*, 6110– 6116, DOI: 10.1063/1.178321230https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXnvFeqs7g%253D&md5=409eb74e4f9716d5bab8ddf7de1e4f8bState-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curveChan, Garnet Kin-Lic; Kallay, Mihaly; Gauss, JurgenJournal of Chemical Physics (2004), 121 (13), 6110-6116CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the nitrogen binding curve with the d. matrix renormalization group (DMRG) and single-ref. and multireference coupled cluster (CC) theory. Our DMRG calcns. use up to 4000 states and our single-ref. CC calcns. include up to full connected hextuple excitations. Using the DMRG, we compute an all-electron benchmark nitrogen binding curve, at the polarized, valence double-zeta level (28 basis functions), with an estd. accuracy of 0.03 mEh. We also assess the performance of more approx. DMRG and CC theories across the nitrogen curve. We provide an anal. of the relative strengths and merits of the DMRG and CC theory under different correlation conditions.**31**Máté, M.; Petrov, K.; Szalay, S.; Legeza, Ö. Compressing multireference character of wave functions via fermionic mode optimization.*J. Math. Chem.*2023,*61*, 362– 375, DOI: 10.1007/s10910-022-01379-y31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XhvFOgs7vJ&md5=ce681e458caee0bd135ea33dd7d87c2cCompressing multireference character of wave functions via fermionic mode optimizationMate, Mihaly; Petrov, Klara; Szalay, Szilard; Legeza, OrsJournal of Mathematical Chemistry (2023), 61 (2), 362-375CODEN: JMCHEG; ISSN:0259-9791. (Springer)Abstr.: In this work, we present a brief overview of the fermionic mode optimization within the framework of tensor network state methods (Krumnow et al. in Phys Rev Lett 117:210402, 2016, https://doi.org/10.1103/PhysRevLett.117.210402), and demonstrate that it has the potential to compress the multireference character of the wave functions after finding optimal MOs (modes), based on entanglement minimization. Numerical simulations have been performed for the nitrogen dimer in the cc-pVDZ basis for the equil. and for stretched geometries.**32**Boguslawski, K.; Tecmer, P. Benchmark of Dynamic Electron Correlation Models for Seniority-Zero Wave Functions and Their Application to Thermochemistry.*J. Chem. Theory Comput.*2017,*13*, 5966– 5983, DOI: 10.1021/acs.jctc.6b0113432https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFWgsrzI&md5=aa12162e85f9aaae838d5117a2ace7d7Benchmark of Dynamic Electron Correlation Models for Seniority-Zero Wave Functions and Their Application to ThermochemistryBoguslawski, Katharina; Tecmer, PawelJournal of Chemical Theory and Computation (2017), 13 (12), 5966-5983CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Wave functions restricted to electron-pair states are promising models to describe static/nondynamic electron correlation effects encountered, for instance, in bond-dissocn. processes and transition-metal and actinide chem. To reach spectroscopic accuracy, however, the missing dynamic electron correlation effects that cannot be described by electron-pair states need to be included a posteriori. In this Article, we extend the previously presented perturbation theory models with an Antisym. Product of 1-ref. orbital Geminal (AP1roG) ref. function that allows us to describe both static/nondynamic and dynamic electron correlation effects. Specifically, our perturbation theory models combine a diagonal and off-diagonal zero-order Hamiltonian, a single-ref. and multireference dual state, and different excitation operators used to construct the projection manifold. We benchmark all proposed models as well as an a posteriori Linearized Coupled Cluster correction on top of AP1roG against CR-CC(2,3) ref. data for reaction energies of several closed-shell mols. that are extrapolated to the basis set limit. Moreover, we test the performance of our new methods for multiple bond breaking processes in the homonuclear N2, C2, and F2 dimers as well as the heteronuclear BN, CO, and CN+ dimers against MRCI-SD, MRCI-SD + Q, and CR-CC(2,3) ref. data. Our numerical results indicate that the best performance is obtained from a Linearized Coupled Cluster correction as well as second-order perturbation theory corrections employing a diagonal and off-diagonal zero-order Hamiltonian and a single-determinant dual state. These dynamic corrections on top of AP1roG provide substantial improvements for binding energies and spectroscopic properties obtained with the AP1roG approach, while allowing us to approach chem. accuracy for reaction energies involving closed-shell species.PMID: 28921980.

**33**Nowak, A.; Legeza, O.; Boguslawski, K. Orbital entanglement and correlation from pCCD-tailored coupled cluster wave functions.*J. Chem. Phys.*2021,*154*, 084111, DOI: 10.1063/5.003820533https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXltFOntL0%253D&md5=e935016c0969f409fbf7ec34903598d7Orbital entanglement and correlation from pCCD-tailored coupled cluster wave functionsNowak, Artur; Legeza, Ors; Boguslawski, KatharinaJournal of Chemical Physics (2021), 154 (8), 084111CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Wave functions based on electron-pair states provide inexpensive and reliable models to describe quantum many-body problems contg. strongly correlated electrons, given that broken-pair states have been appropriately accounted for by, for instance, a posteriori corrections. In this article, we analyze the performance of electron-pair methods in predicting orbital-based correlation spectra. We focus on the (orbital-optimized) pair-coupled cluster doubles (pCCD) ansatz with a linearized coupled-cluster (LCC) correction. Specifically, we scrutinize how orbital-based entanglement and correlation measures can be detd. from a pCCD-tailored CC wave function. Furthermore, we employ the single-orbital entropy, the orbital-pair mutual information, and the eigenvalue spectra of the two-orbital reduced d. matrixes to benchmark the performance of the LCC correction for the one-dimensional Hubbard model with the periodic boundary condition as well as the N2 and F2 mols. against d. matrix renormalization group ref. calcns. Our study indicates that pCCD-LCC accurately reproduces the orbital-pair correlation patterns in the weak correlation limit and for mols. close to their equil. structure. Hence, we can conclude that pCCD-LCC predicts reliable wave functions in this regime. (c) 2021 American Institute of Physics.**34**Leszczyk, A.; Máté, M.; Legeza, O.; Boguslawski, K. Assessing the Accuracy of Tailored Coupled Cluster Methods Corrected by Electronic Wave Functions of Polynomial Cost.*J. Chem. Theory Comput.*2022,*18*, 96– 117, DOI: 10.1021/acs.jctc.1c0028434https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXivVeksrvN&md5=5785b5dbc6bb45629aa9eb30c5a00797Assessing the Accuracy of Tailored Coupled Cluster Methods Corrected by Electronic Wave Functions of Polynomial CostLeszczyk, Aleksandra; Mate, Mihaly; Legeza, Ors; Boguslawski, KatharinaJournal of Chemical Theory and Computation (2022), 18 (1), 96-117CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Tailored coupled cluster theory represents a computationally inexpensive way to describe static and dynamical electron correlation effects. In this work, we scrutinize the performance of various coupled cluster methods tailored by electronic wave functions of polynomial cost. Specifically, we focus on frozen-pair coupled cluster (fpCC) methods, which are tailored by pair-coupled cluster doubles (pCCD), and coupled cluster theory tailored by matrix product state wave functions optimized by the d. matrix renormalization group (DMRG) algorithm. As test system, we selected a set of various small- and medium-sized mols. contg. diatomics (N2, F2, C2, CN+, CO, BN, BO+, and Cr2) and mols. (ammonia, ethylene, cyclobutadiene, benzene, hydrogen chains, rings, and cuboids) for which the conventional single-ref. coupled cluster singles and doubles (CCSD) method is not able to produce accurate results for spectroscopic consts., potential energy surfaces, and barrier heights. Most importantly, DMRG-tailored and pCCD-tailored approaches yield similar errors in spectroscopic consts. and potential energy surfaces compared to accurate theor. and/or exptl. ref. data. Although fpCC methods provide a reliable description for the dissocn. pathway of mols. featuring single and quadruple bonds, they fail in the description of triple or hextuple bond-breaking processes or avoided crossing regions.**35**Nowak, A.; Boguslawski, K. A configuration interaction correction on top of pair coupled cluster doubles.*Phys. Chem. Chem. Phys.*2023,*25*, 7289– 7301, DOI: 10.1039/D2CP05171K35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3sXjs1emsLo%253D&md5=a82af0b8ba3799cb795584540a76a282A configuration interaction correction on top of pair coupled cluster doublesNowak, Artur; Boguslawski, KatharinaPhysical Chemistry Chemical Physics (2023), 25 (10), 7289-7301CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Numerous numerical studies have shown that geminal-based methods are a promising direction to model strongly correlated systems with low computational costs. Several strategies have been introduced to capture the missing dynamical correlation effects, which typically exploit a posteriori corrections to account for correlation effects assocd. with broken-pair states or inter-geminal correlations. In this article, we scrutinize the accuracy of the pair coupled cluster doubles (pCCD) method extended by CI (CI) theory. Specifically, we benchmark various CI models, including, at most double excitations against selected CC corrections as well as conventional single-ref. CC methods. A simple Davidson correction is also tested. The accuracy of the proposed pCCD-CI approaches is assessed for challenging small model systems such as the N2 and F2 dimers and various di- and triat. actinide-contg. compds. In general, the proposed CI methods considerably improve spectroscopic consts. compared to the conventional CCSD approach, provided a Davidson correction is included in the theor. model. At the same time, their accuracy lies between those of the linearized frozen pCCD and frozen pCCD variants.**36**Li, X.; Paldus, J. Dissociation of N2 triple bond: a reduced multireference CCSD study.*Chem. Phys. Lett.*1998,*286*, 145– 154, DOI: 10.1016/S0009-2614(97)01132-936https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXhvF2isLY%253D&md5=1903dac9e4922e561f848ffc7f2472daDissociation of N2 triple bond: a reduced multireference CCSD studyLi, Xiangzhu; Paldus, JosefChemical Physics Letters (1998), 286 (1,2), 145-154CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)The reduced multireference coupled cluster method with singles and doubles is applied to the dissocn. of the ground state of the N mol. Even with a relatively modest highly truncated ref. space one obtains an accurate potential over a wide range of internuclear sepns.**37**Leszczyk, A.; Dome, T.; Tecmer, P.; Kedziera, D.; Boguslawski, K. Resolving the π-assisted U–N σf-bond formation using quantum information theory.*Phys. Chem. Chem. Phys.*2022,*24*, 21296– 21307, DOI: 10.1039/D2CP03377AThere is no corresponding record for this reference.**38**Kramers, H. A. Théorie générale de la rotation paramagnétique dans les cristaux.*Proceedings of the Royal Netherlands Academy of Arts and Sciences*1930,*33*(6-10), 959– 972There is no corresponding record for this reference.**39**Fleig, T.; Olsen, J.; Marian, C. M. The generalized active space concept for the relativistic treatment of electron correlation. I. Kramers-restricted two-component configuration interaction.*J. Chem. Phys.*2001,*114*, 4775– 4790, DOI: 10.1063/1.134907639https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXhslWisLo%253D&md5=406052d6433f64aec91b79c33cd878a8The generalized active space concept for the relativistic treatment of electron correlation. I. Kramers-restricted two-component configuration interactionFleig, Timo; Olsen, Jeppe; Marian, Christel M.Journal of Chemical Physics (2001), 114 (11), 4775-4790CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)As a prelude to a series of presentations dealing with the treatment of electron correlation and special relativity, we present the theor. background and the implementation of a new two-component relativistic CI program. It is based on the method of generalized active spaces which has been extended from a nonrelativistic implementation to make use of two-component Hamiltonians and time reversal and double point group symmetry at both the spinor and Slater determinant level. We demonstrate how the great computational effort arising from such a general approach-the treatment of spin-orbit interaction and electron correlation in a fully variational framework-can be markedly reduced by the use of the aforementioned symmetries. Evidence for the performance of the program is given through a no. of calcns. on light systems with a significant spin-orbit splitting in low-lying electronic states and the well-known problem case thallium, which often serves as a rigorous test system in relativistic electronic structure calcns.**40**Saue, T. DIRAC, a relativistic ab initio electronic structure program, 2018. http://www.diracprogram.org.There is no corresponding record for this reference.**41**Visscher, L.*DIRAC24*, 2024. https://zenodo.org/doi/10.5281/zenodo.10680560.There is no corresponding record for this reference.**42**Saue, T.; Jensen, H. J. A. Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method.*J. Chem. Phys.*1999,*111*, 6211– 6222, DOI: 10.1063/1.47995842https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXmtFKrsbk%253D&md5=54f0466503e395c70c97ab7df3c96ea8Quaternion symmetry in relativistic molecular calculations: the Dirac-Hartree-Fock methodSaue, T.; Jensen, H. J. AaJournal of Chemical Physics (1999), 111 (14), 6211-6222CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A symmetry scheme based on the irreducible co-representations of the full symmetry group of a mol. system is presented for use in relativistic calcns. Consideration of time-reversal symmetry leads to a reformulation of the Dirac-Hartree-Fock equations in terms of quaternion algebra. Further symmetry redns. due to mol. point group symmetry are then manifested by a descent to complex or real algebra. Spatial symmetry is restricted to D2h and subgroups, and it is demonstrated that the Frobenius-Schur test can be used to characterize these groups as a whole. The resulting symmetry scheme automatically provides max. point group and time-reversal symmetry redn. of the computational effort, also when the Fock matrix is constructed in a scalar basis, i.e., from the same type of electron repulsion integrals over symmetry-adapted scalar basis functions as in nonrelativistic theory. An illustrative numerical example is given showing symmetry redns. comparable to the nonrelativistic case.**43**Dyall, K. G.; Fægri, K., Jr.*Introduction to Relativistic Quantum Chemistry*; Oxford University Press, 2007.There is no corresponding record for this reference.**44**Visscher, L. On the construction of double group molecular symmetry functions.*Chem. Phys. Lett.*1996,*253*, 20– 26, DOI: 10.1016/0009-2614(96)00234-5There is no corresponding record for this reference.**45**Visscher, L. The Dirac equation in quantum chemistry: Strategies to overcome the current computational problems.*J. Comput. Chem.*2002,*23*, 759– 766, DOI: 10.1002/jcc.1003645https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xjs1Wnsb0%253D&md5=170d5cd0fdc7b2347067c3cad6d4ce6bThe Dirac equation in quantum chemistry: strategies to overcome the current computational problemsVisscher, LucasJournal of Computational Chemistry (2002), 23 (8), 759-766CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A perspective on the use of the relativistic Dirac equation in quantum chem. is given. It is demonstrated that many of the computational problems that plague the current implementations of the different electronic structure methods can be overcome by utilizing the locality of the small component wave function and d. Possible applications of such new and more efficient formulations are discussed.**46**Thyssen, J. Development and Applications of Methods for Correlated Relativistic Calculations of Molecular Properties. Ph.D. thesis, University of Southern Denmark, 2001.There is no corresponding record for this reference.**47**Li, X.; Paldus, J. Reduced multireference CCSD method: An effective approach to quasidegenerate states.*J. Chem. Phys.*1997,*107*, 6257, DOI: 10.1063/1.47428947https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXmsFOhtro%253D&md5=58247434ded7440fd6a789c3f054c993Reduced multireference CCSD method: an effective approach to quasidegenerate statesLi, Xiangzhu; Paldus, JosefJournal of Chemical Physics (1997), 107 (16), 6257-6269CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Std. multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple soln. problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that conc. on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) the construction of a variational CI wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster anal. of this CI wave function providing the information about the higher than pair cluster amplitudes, and (iii) the exploitation of these amplitudes in the so-called externally cor. CCSD procedure. This approach is referred to as the reduced MR (RMR) SS CCSD method and is implemented at the ab initio level and applied to several model systems for which the exact full CI results are available. These include two four electron H4 systems (usually referred to as the H4 and S4 models), an eight electron H8 model, and the singlet-triplet sepn. problem in CH2. It is shown that the RMR CCSD approach produces highly accurate results, is free from intruder state problems, is very general and effective and applicable to both closed and open shell systems.**48**Lyakh, D. I.; Lotrich, V. F.; Bartlett, R. J. The tailored CCSD(T) description of the automerization of cyclobutadiene.*Chem. Phys. Lett.*2011,*501*, 166– 171, DOI: 10.1016/j.cplett.2010.11.05848https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXjvVWg&md5=376e9113d921f85860cf34467b3e685eThe 'tailored' CCSD(T) description of the automerization of cyclobutadieneLyakh, Dmitry I.; Lotrich, Victor F.; Bartlett, Rodney J.Chemical Physics Letters (2011), 501 (4-6), 166-171CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)An alternative route to extend the CCSD(T) approach to multireference problems is presented. The well-known defect of the CCSD(T) model in describing the non-dynamic electron correlation effects is remedied by 'tailoring' the underlying coupled-cluster singles and doubles (CCSD) approach and applying the perturbative triples correction to it. The TCCSD(T) approach suggested in the paper has the same computational demands as the CCSD(T) method, though being mostly free from its drawbacks pertinent to multireference (quasidegenerate) situations. To test the approach we calc. the potential energy surface for the automerization of cyclobutadiene where the transition state exhibits a strong multireference character.**49**Melnichuk, A.; Bartlett, R. J. Relaxed active space: Fixing tailored-CC with high order coupled cluster. I.*J. Chem. Phys.*2012,*137*, 214103, DOI: 10.1063/1.476790049https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhslOrsrfL&md5=1a4df0dd207e744a42cd87f983feaf18Relaxed active space: Fixing tailored-CC with high order coupled cluster. IMelnichuk, Anna; Bartlett, Rodney J.Journal of Chemical Physics (2012), 137 (21), 214103/1-214103/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Several single ref. (SR-CC) coupled cluster methods are shown to work for traditionally multi-ref. (MR) problems such as bond breaking subject to RHF refs. The correlated methods can successfully describe any MR problem with enough higher order clusters: singles and doubles (CCSD), singles, doubles and triples (CCSDT), singles, doubles, triples, and quadruples (CCSDTQ), etc. However, due to the steep increase in the computational cost, it is not practical to do larger systems or to use large basis sets without active space partitioning. In this study, the orbital space is partitioned into an active space subject to an unambiguous statistical criteria to span the MR behavior which defines an extended space to let the active space relax. The rest is considered the external space. The extended space is treated with CCSDT and the external space with CCSD. An automated scheme for detg. the extended space is presented and evaluated. We build upon the tailored-CC scheme of Hino and address its main shortcoming of neglecting the coupling between the active space and the rest of the orbital space which results in loss of accuracy as well as a pronounced nonparallelism error (NPE). The automated scheme makes it unnecessary for the user to judge whether a chosen active space is sufficient to correctly solve the problem. We illustrate this method for the hydrogen fluoride and fluorine mol. ground state dissocn. potentials using the extended space partitioning methods. Exptl. accuracy for the dissocn. energy is achieved at a small fraction of the cost of doing a full CCSDT calcn. This approach is easily amendable to higher order clusters which are required for double and triple bond breaking and other strongly multi-ref. systems. (c) 2012 American Institute of Physics.**50**Melnichuk, A.; Bartlett, R. J. Relaxed active space: Fixing tailored-CC with high order coupled cluster. II.*J. Chem. Phys.*2014,*140*, 064113, DOI: 10.1063/1.486267650https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXjvFyms7k%253D&md5=c67b98a2d4ecfaed095330cf6cdc36fbRelaxed active space: Fixing tailored-CC with high order coupled cluster. IIMelnichuk, Ann; Bartlett, Rodney J.Journal of Chemical Physics (2014), 140 (6), 064113/1-064113/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Due to the steep increase in computational cost with the inclusion of higher-connected cluster operators in coupled-cluster applications, it is usually not practical to use such methods for larger systems or basis sets without an active space partitioning. This study generates an active space subject to unambiguous statistical criteria to define a space whose size permits treatment at the CCSDT level. The automated scheme makes it unnecessary for the user to judge whether a chosen active space is sufficient to correctly solve the problem. Two demanding applications are presented: twisted ethylene and the transition states for the bicyclo[1,1,0]butane isomerization. As bi-radicals both systems require at least a CCSDT level of theory for quant. results, for the geometries and energies. (c) 2014 American Institute of Physics.**51**Piecuch, P.; Oliphant, N.; Adamowicz, L. A state-selective multireference coupled-cluster theory employing the single-reference formalism.*J. Chem. Phys.*1993,*99*, 1875– 1900, DOI: 10.1063/1.46617951https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtFWjsrc%253D&md5=85bd8b4c37a2e1763e9032287669e1e5A state-selective multireference coupled-cluster theory employing the single-reference formalismPiecuch, Piotr; Oliphant, Nevin; Adamowicz, LudwikJournal of Chemical Physics (1993), 99 (3), 1875-900CODEN: JCPSA6; ISSN:0021-9606.A new state-selective multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism is described. It is an extension of a simple two-ref. formalism, which the authors presented in the authors' earlier paper [N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991)], and a rigorous formulation of another method of ours, which the authors obtained as an approxn. of the SRCC approach truncated at triple excitations (SRCCSDT) [N. Oliphant and L. Adamowicz, J. Chem. Phys. 96, 3739 (1992)]. The size extensivity of the resulting correlation energies is achieved by employing a SRCC-like ansatz for the multideterminantal wave function. General considerations are supplemented by suggesting a hierarchy of approx. schemes, with the MRCCSD approach (MRCC approach truncated at double excitations from the ref. determinants) representing the most important one. The authors' state-selective MRCCSD theory emerges through a suitable selection of the most essential cluster components appearing in the full SRCCSDTQ method (SRCC method truncated at quadruple excitations), when the latter is applied to quasidegenerate states. The complete set of equations describing the authors' MRCCSD formalism is presented and the possibility of the recursive intermediate factorization [S. A. Kucharski and R. J. Bartlett, Theor. Chim. Acta 80, 387 (1991)] of the authors' approach, leading to an efficient computer algorithm, is discussed.**52**Piecuch, P.; Adamowicz, L. State-selective multireference coupled-cluster theory employing the single-reference formalism: Implementation and application to the H8 model system.*J. Chem. Phys.*1994,*100*, 5792– 5809, DOI: 10.1063/1.46714352https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXjtVOksrY%253D&md5=b8df2b4213996bf687d716f91ccaa240State-selective multireference coupled-cluster theory employing the single-reference formalism: implementation and application to the H8 model systemPiecuch, Piotr; Adamowicz, LudwikJournal of Chemical Physics (1994), 100 (8), 5792-809CODEN: JCPSA6; ISSN:0021-9606.The new state-selective (SS) multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism, which was introduced previously (P. Piecuch, et al., 1993), was implemented; results are presented of pilot calcns. for the min. basis-set (MBS) model composed of eight hydrogen atoms in various geometrical arrangements. This model enables a continuous transition between degenerate and nondegenerate regimes. Comparison is made with the results of SR CC calcns. involving double (CCD), single and double (CCSD), single, double, and triple (CCSDT), and single, double, triple, and quadruple (CCSDTQ) excitations. The authors' SS CC energies are also compared with the results of Hilbert space, state-universal (SU) MR CC(S)D calcns., as well as with MR-CI results (with and without Davidson-type corrections), and with exact correlation energies obtained using the full-CI (FCI) method. Along with the ground-state energies, the authors also analyzed the resulting wave functions by examg. some selected cluster components. This anal. enabled the authors to assess the quality of the resulting wave functions. The authors' SS CC theory truncated at double excitations, which emerges through selection of the most essential clusters appearing in the full SR CCSDTQ formalism [SS CCSD (TQ) method], provided equally good results in the nondegenerate and quasidegenerate regions. The difference between the ground-state energy obtained with the SS CCSD(TQ) approach and the FCI energy did not exceed 1.1 milli-hartree over all the geometries considered. This value compares favorably with the max. difference of 2.8 milli-hartrees between the SU CCSD energies and the FCI energies obtained for the same range of geometries. The SS CCSD(T) method, emerging from the SR CCSDT theory through selection of the most essential clusters, was less stable, since it neglected very important semi-internal quadruple excitations. Unlike the genuine multideterminantal SU CC formalism, the authors' SS CC approach was not affected by the intruder-state problem, and its convergence remained satisfactory the in nondegenerate and quasidegenerate regimes.**53**Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy.*Phys. Rev. B*2003,*68*, 195116, DOI: 10.1103/PhysRevB.68.19511653https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXpvVegs7k%253D&md5=215685d20c465a36d96e9adf4bbb0ea3Optimizing the density-matrix renormalization group method using quantum information entropyLegeza, O.; Solyom, J.Physical Review B: Condensed Matter and Materials Physics (2003), 68 (19), 195116/1-195116/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)In order to optimize the ordering of the lattice sites in the momentum space and quantum chem. versions of the d.-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various mols. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chem. DMRG to a great extent. The effect of lattice site ordering on the no. of block states to be kept during the RG procedure is also investigated.**54**Battaglia, S.; Keller, S.; Knecht, S. Efficient Relativistic Density-Matrix Renormalization Group Implementation in a Matrix-Product Formulation.*J. Chem. Theory Comput.*2018,*14*, 2353– 2369, DOI: 10.1021/acs.jctc.7b0106554https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXltFens78%253D&md5=c9dc1a06d66b433bfa2dcbc1a902d153Efficient Relativistic Density-Matrix Renormalization Group Implementation in a Matrix-Product FormulationBattaglia, Stefano; Keller, Sebastian; Knecht, StefanJournal of Chemical Theory and Computation (2018), 14 (5), 2353-2369CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present an implementation of the relativistic quantum-chem. d. matrix renormalization group (DMRG) approach based on a matrix-product formalism. Our approach allows us to optimize matrix product state (MPS) wave functions including a variational description of scalar-relativistic effects and spin-orbit coupling from which we can calc., for example, first-order elec. and magnetic properties in a relativistic framework. While complementing our pilot implementation (Knecht, S. et al., J. Chem. Phys. 2014, 140, 041101), this work exploits all features provided by its underlying nonrelativistic DMRG implementation based on an matrix product state and operator formalism. We illustrate the capabilities of our relativistic DMRG approach by studying the ground-state magnetization, as well as c.d. of a paramagnetic f9 dysprosium complex as a function of the active orbital space employed in the MPS wave function optimization.**55**Szalay, S.; Pfeffer, M.; Murg, V.; Barcza, G.; Verstraete, F.; Schneider, R.; Legeza, Ö. Tensor product methods and entanglement optimization forab initioquantum chemistry.*Int. J. Quantum Chem.*2015,*115*, 1342– 1391, DOI: 10.1002/qua.2489855https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXovVags7c%253D&md5=d1f7db2c2c73a2d907e9a290ccb7f22bTensor product methods and entanglement optimization for ab initio quantum chemistrySzalay, Szilard; Pfeffer, Max; Murg, Valentin; Barcza, Gergely; Verstraete, Frank; Schneider, Reinhold; Legeza, OersInternational Journal of Quantum Chemistry (2015), 115 (19), 1342-1391CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The treatment of high-dimensional problems such as the Schroedinger equation can be approached by concepts of tensor product approxn. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods-developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools-can be combined to attack highly challenging problems in quantum chem. The aim of the present paper is to give a pedagogical introduction to the theor. background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum chem." at the Workshop on Theor. Chem., Feb. 18-21, 2014, Mariapfarr, Austria. © 2015 Wiley Periodicals, Inc.**56**Menczer, A.; Legeza, Ö. Massively Parallel Tensor Network State Algorithms on Hybrid CPU-GPU Based Architectures.*arXiv*2023, arXiv:2305.05581v1. DOI: 10.48550/arXiv.2305.05581 .There is no corresponding record for this reference.**57**Menczer, A.; Legeza, Ö. Boosting the effective performance of massively parallel tensor network state algorithms on hybrid CPU-GPU based architectures via non-Abelian symmetries, 2023. https://arxiv.org/abs/2309.16724.There is no corresponding record for this reference.**58**Menczer, A.; Kapás, K.; Werner, M. A.; Legeza, O. Two-dimensional quantum lattice models via mode optimized hybrid CPU-GPU density matrix renormalization group method.*Phys. Rev. B*2024,*109*, 195148, DOI: 10.1103/PhysRevB.109.195148There is no corresponding record for this reference.**59**Menczer, A.; van Damme, M.; Rask, A.; Huntington, L.; Hammond, J.; Xantheas, S. S.; Ganahl, M.; Legeza, Ö. Parallel implementation of the Density Matrix Renormalization Group method achieving a quarter petaFLOPS performance on a single DGX-H100 GPU node.*J. Chem. Theory Comput.*2024,*20*, 8397– 8404, DOI: 10.1021/acs.jctc.4c00903There is no corresponding record for this reference.**60**Brandejs, J.; Pototschnig, J.; Saue, T. Generating coupled cluster code for modern distributed memory tensor software, 2024. https://arxiv.org/abs/2409.06759.There is no corresponding record for this reference.**61**Atkinson, B. E.; Hu, H.-S.; Kaltsoyannis, N. Post Hartree–Fock calculations of pnictogen–uranium bonding in EUF3 (E = N–Bi).*Chem. Commun.*2018,*54*, 11100– 11103, DOI: 10.1039/C8CC05581EThere is no corresponding record for this reference.**62**TURBOMOLE, v7.52020, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH,1989–2007. https://www.turbomole.org.There is no corresponding record for this reference.**63**Dyall, K. G. Core correlating basis functions for elements 31–118.*Theor. Chem. Acc.*2012,*131*, 1217, DOI: 10.1007/s00214-012-1217-8There is no corresponding record for this reference.**64**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: 10.1063/1.45615364https://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.**65**Dunning, T. H., Jr. ANL vibration–rotation analysis program for diatomic molecules, 1979.There is no corresponding record for this reference.**66**Visscher, L. Approximate molecular Dirac-Coulomb calculations using a simple Coulombic correction.*Theor. Chem. Acc.*1997,*98*, 68, DOI: 10.1007/s00214005028066https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXotFamu74%253D&md5=ac65458dcfe7cc47f433e92757371b92Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correctionVisscher, LucasTheoretical Chemistry Accounts (1997), 98 (2-3), 68-70CODEN: TCACFW; ISSN:1432-881X. (Springer-Verlag)A simple point-charge model is used to correct mol. 4-component Dirac-Coulomb calcns. which neglect 2-electron integrals over the small components of the wave function. The calcd. valence properties show no degeneration relative to the full calcn., while a speed-up factor of 3 is obtained.**67**Andrews, L.; Wang, X.; Lindh, R.; Roos, B.; Marsden, C. Simple NUF3 and PUF3Molecules with Triple Bonds to Uranium.*Angew. Chem., Int. Ed.*2008,*47*, 5366– 5370, DOI: 10.1002/anie.200801120There is no corresponding record for this reference.**68**Van Gundy, R. A. Electronic Structure of Metal-Containing Diatomic Ions. Ph.D. thesis, Faculty of the James T. Laney School of Graduate Studies of Emory University, 2018.There is no corresponding record for this reference.**69**King, D. M.; Liddle, S. T. Progress in molecular uranium-nitride chemistry.*Coord. Chem. Rev.*2014,*266–267*, 2– 15, DOI: 10.1016/j.ccr.2013.06.01369https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXjsVChtrc%253D&md5=ddd0c7bd4cb450c7582ebd60bea3fe05Progress in molecular uranium-nitride chemistryKing, David M.; Liddle, Stephen T.Coordination Chemistry Reviews (2014), 266-267 (), 2-15CODEN: CCHRAM; ISSN:0010-8545. (Elsevier B.V.)A review. The coordination, organometallic, and materials chem. of uranium nitride has long been an important facet of actinide chem. Following matrix isolation expts. and computational characterization, mol., soln.-based uranium chem. has developed significantly in the last decade or so culminating most recently in the isolation of the first examples of long-sought terminal uranium nitride linkages. Herein, the field is reviewed with an emphasis on well-defined mol. species.**70**Balasubramanian, S. G. TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations.*J. Chem. Phys.*2020,*152*, 184107, DOI: 10.1063/5.0004635There is no corresponding record for this reference.**71**Furness, J. W.; Kaplan, A. D.; Ning, J.; Perdew, J. P.; Sun, J. Accurate and Numerically Efficient r2SCAN Meta-Generalized Gradient Approximation.*J. Phys. Chem. Lett.*2020,*11*, 8208– 8215, DOI: 10.1021/acs.jpclett.0c0240571https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhslequ77N&md5=49adb31d49e1e53910d87275f6400ae9Accurate and Numerically Efficient r2SCAN Meta-Generalized Gradient ApproximationFurness, James W.; Kaplan, Aaron D.; Ning, Jinliang; Perdew, John P.; Sun, JianweiJournal of Physical Chemistry Letters (2020), 11 (19), 8208-8215CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)The recently proposed rSCAN functional [J. Chem. Phys., 2019, 150, 161101] is a regularized form of the SCAN functional [Phys. Rev. Lett., 2015, 115, 036402] that improves SCAN's numerical performance at the expense of breaking constraints known from the exact exchange-correlation functional. We construct a new meta-generalized gradient approxn. by restoring exact constraint adherence to rSCAN. The resulting functional maintains rSCAN's numerical performance while restoring the transferable accuracy of SCAN.**72**Holzer, C.; Franzke, Y. J.; Kehry, M. Assessing the Accuracy of Local Hybrid Density Functional Approximations for Molecular Response Properties.*J. Chem. Theory Comput.*2021,*17*, 2928– 2947, DOI: 10.1021/acs.jctc.1c0020372https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXpslejsrY%253D&md5=9a7b4bbcbd5a5b3dd35fce208f203513Assessing the Accuracy of Local Hybrid Density Functional Approximations for Molecular Response PropertiesHolzer, Christof; Franzke, Yannick J.; Kehry, MaxJournal of Chemical Theory and Computation (2021), 17 (5), 2928-2947CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A comprehensive overview of the performance of local hybrid functionals for mol. properties like excited states, ionization potentials within the GW framework, polarizabilities, magnetizabilities, NMR chem. shifts, and NMR spin-spin coupling consts. is presented. We apply the generalization of the kinetic energy, τ, with the paramagnetic c.d. to all magnetic properties and the excitation energies from time-dependent d. functional theory. This restores gauge invariance for these properties. Different ansatze for local mixing functions such as the iso-orbital indicator, the correlation length, the Gorling-Levy second-order limit, and the spin polarization are compared. For the latter, we propose a modified version of the corresponding hyper-generalized gradient approxn. functional of Perdew, Staroverov, Tao, and Scuseria (PSTS) to allow for a numerically stable evaluation of the exchange-correlation kernel and hyperkernel. The PSTS functional leads to a very consistent improvement compared to the related TPSSh functional. It is further shown that the "best" choice of the local mixing function depends on the studied property and mol. class. While functionals based on the iso-orbital indicator lead to rather accurate excitation energies and ionization energies, the results are less impressive for NMR properties, for which a considerable dependence on the considered mol. test set and nuclei is obsd. Johnson's local hybrid functional based on the correlation length yields remarkable results for NMR shifts of compds. featuring heavy elements and also for the excitation energies of org. compds.**73**Perdew, J. P.; Staroverov, V. N.; Tao, J.; Scuseria, G. E. Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfaction.*Phys. Rev. A*2008,*78*, 052513, DOI: 10.1103/PhysRevA.78.05251373https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhsVGqtbbP&md5=4b1748d8e8d9deb01f650564d60ec7e5Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfactionPerdew, John P.; Staroverov, Viktor N.; Tao, Jianmin; Scuseria, Gustavo E.Physical Review A: Atomic, Molecular, and Optical Physics (2008), 78 (5, Pt. A), 052513/1-052513/13CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We construct a nonlocal d. functional approxn. with full exact exchange, while preserving the constraint-satisfaction approach and justified error cancellations of simpler semilocal functionals. This is achieved by interpolating between different approxns. suitable for two extreme regions of the electron d. In a "normal" region, the exact exchange-correlation hole d. around an electron is semilocal because its spatial range is reduced by correlation and because it integrates over a narrow range to -1. These regions are well described by popular semilocal approxns. (many of which have been constructed nonempirically), because of proper accuracy for a slowly varying d. or because of error cancellation between exchange and correlation. "Abnormal" regions, where nonlocality is unveiled, include those in which exchange can dominate correlation (one-electron, nonuniform high d., and rapidly varying limits), and those open subsystems of fluctuating electron no. over which the exact exchange-correlation hole integrates to a value greater than -1. Regions between these extremes are described by a hybrid functional mixing exact and semilocal exchange energy densities locally, i.e., with a mixing fraction that is a function of position r and a functional of the d. Because our mixing fraction tends to 1 in the high-d. limit, we employ full exact exchange according to the rigorous definition of the exchange component of any exchange-correlation energy functional. Use of full exact exchange permits the satisfaction of many exact constraints, but the nonlocality of exchange also requires balanced nonlocality of correlation. We find that this nonlocality can demand at least five empirical parameters, corresponding roughly to the four kinds of abnormal regions. Our local hybrid functional is perhaps the first accurate fourth-rung d. functional or hyper-generalized gradient approxn., with full exact exchange, that is size-consistent in the way that simpler functionals are. It satisfies other known exact constraints, including exactness for all one-electron densities, and provides an excellent fit to the 223 mol. enthalpies of formation of the G3/99 set and the 42 reaction barrier heights of the BH42/03 set, improving both (but esp. the latter) over most semilocal functionals and global hybrids. Exact constraints, phys. insights, and paradigm examples hopefully suppress "overfitting.".

## Supporting Information

## Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.4c00641.

Files for frequency recomputation (ZIP)

Optimized geometries, assessment of XC functionals, ECP computational details, visualization of active molecular orbitals, TCC active space optimization, results in MP2 NO basis, orbital entropy figures (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.