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Simplified State Interaction for Matrix Product State Wave Functions

Cite this: J. Chem. Theory Comput. 2021, 17, 12, 7477–7485
Publication Date (Web):December 3, 2021
https://doi.org/10.1021/acs.jctc.1c00674

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Abstract

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We present an approximation to the state-interaction approach for matrix product state (MPS) wave functions (MPSSI) in a nonorthogonal molecular orbital basis, first presented by Knecht et al. [J. Chem. Theory Comput.,2016, 28, 5881], that allows for a significant reduction of the computational cost without significantly compromising its accuracy. The approximation is well-suited if the molecular orbital basis is close to orthogonality, and its reliability may be estimated a priori with a single numerical parameter. For an example of a platinum azide complex, our approximation offers up to 63-fold reduction in computational time compared to the original method for wave function overlaps and spin–orbit couplings, while still maintaining numerical accuracy.

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1. Introduction

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Accurate calculations of many photochemical processes can be a daunting task. Excited states are often governed by strong electron correlation effects and many close-lying excited states, where multiconfigurational electronic structure methods (1,2) are indispensable.
Multiconfigurational methods based on the complete active space self-consistent field (CASSCF) (3) are well-established for handling strong correlation effects. These approaches require selecting an orbital subspace called active orbital space whose size determines the computational cost. Traditional CASSCF methods scale exponentially with the number of the active orbitals and electrons, allowing for calculations of up to 22 electrons in 22 orbitals with a massively parallel approach (4) but limiting its size to approximately 18 electrons in 18 orbitals (5) under more moderate computational time requirements. These limits can be reached very quickly, especially in polynuclear transition-metal complexes. One approach to overcome the exponential scaling of CASSCF is the density matrix renormalization group (DMRG) (6,7) for quantum chemistry, (8−16) which, combined with self-consistent field orbital optimization (DMRG-SCF), (17,18) is able to variationally approximate CASSCF wave functions to arbitrary accuracy at a polynomial instead of exponential scaling of the computational cost.
In the CASSCF paradigm, and also with DMRG-SCF, excited states are usually calculated with a state-average ansatz, where a single orthonormal set of molecular orbitals (MOs) is optimized to provide a balanced representation of several states. This allows for a straightforward calculation of transition densities and moments that are required to compute properties such as oscillator strengths, magnetic properties, or spin–orbit couplings. However, state-averaging is not always possible or desired: (i) the individual state characters differ too much for an average set of orbitals to yield an adequate description; (ii) state-averaging, for example, between different spin multiplicities, is not supported by the computer implementation of the method, or (iii) a single molecular set of orbitals is simply not possible at all. The latter problem is encountered, for instance, when calculating the overlap between wave functions that are associated with different molecular structures to monitor the change in the character of the electronic wave function, as described in ref (19). In such cases, each state is optimized independently and the resulting MO bases for the individual states are no longer the same. As a consequence, the states are no longer mutually orthogonal, turning the calculation of transition densities and moments into a challenging task.
A solution to this predicament is to use the complete active space state interaction (CASSI) method, proposed by Malmqvist and Roos, (20,21) who suggested to transform the MO bases for the individual states to a biorthonormal basis. Along with the orbital rotation, this requires a simultaneous “counter-rotation” of the wave function expansion coefficients: for a configuration interaction (CI)-type wave functions, which include CASSCF wave functions, this step can be achieved with a series of single-orbital transformations. (21,22) After transformation to biorthonormal basis, wave function overlaps and transition densities may be evaluated at little to no computational overhead.
The CASSI approach was soon extended to calculate spin–orbit couplings, (23) and with the advent of DMRG for quantum chemistry, the DMRG-based version of CASSI, later named the matrix product state (MPS) state interaction (sic!) (MPSSI), has been introduced. (24) To account for spin–orbit interaction with DMRG wave functions, several other approaches based on spin-free wave functions that share a common MO basis were developed (25−27) as well as a fully relativistic four-component approach. (28,29)
Multiconfigurational methods and, specifically, the CASSCF method are often used as an underlying method for the electronic structure calculations of ab initio nonadiabatic dynamics: (30) partially due to their computational efficiency for small systems and ability to describe strong correlation but also because of the readily available implementations for gradients and nonadiabatic couplings. (31−34) With the help of the CASSI method, ab initio nonadiabatic excited-state dynamics with spin–orbit couplings, for example, with the SHARC approach, (35) may be employed to study processes involving different spin states coupled via intersystem crossing. Additionally, overlaps between wave functions at different time steps may also be calculated with CASSI and may serve to approximate nonadiabatic couplings that are included in the on-the-fly propagation of nuclear wave functions. (36) The steep scaling of CASSCF with respect to the active orbital space may be tamed with DMRG-SCF also with surface-hopping dynamics, as the calculation of the analytical gradients and nonadiabatic couplings has been recently reported, (37) and spin–orbit couplings and wave function overlaps may be calculated with MPSSI. DMRG-SCF, despite its polynomial scaling of the computational time with the active space size, is, nevertheless, computationally very intensive, and MPSSI is also a cost-intensive method with a computational cost comparable to that of DMRG-SCF. Dynamics calculations, however, require a cheap and performant electronic structure method, as electronic structure calculations of energies, gradients, and couplings are carried out for hundreds or thousands of time steps. Accordingly, elimination of every possible bottleneck in the electronic structure calculations is extremely beneficial for dynamics calculations.
With the aim of making DMRG broadly applicable to ab initio molecular dynamics simulations, in the present work, we identify the main computational bottlenecks of an MPSSI calculation. Then, we extensively benchmark the sensitivity of the MPSSI accuracy to the choice of the simulation parameters and identify the simulation setup that yields the best compromise between computational cost and accuracy. This optimal setup relies on two approximations, that is, a simple-yet-effective implementation of the orbital rotation operator and an efficient MPS truncation scheme. The error introduced in these two steps is controlled by a single parameter that can, therefore, be tuned based on the target simulation accuracy. We demonstrate the effectiveness of these approximations by means of MPSSI calculations of wave function overlaps and spin–orbit couplings for a medium-sized transition-metal complex.

2. Theory

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As the starting points of this work, we first outline the CASSI and the MPSSI approaches. We assume two sets of multiconfigurational wave functions |ΨX⟩ and |ΨY⟩, each expressed in their own MO basis {ϕpX} and {ϕpY}, respectively, which are not mutually orthogonal. The goal of the CASSI approach is to find the biorthonormal MO bases {ϕpA} and {ϕpB} such that
(1)
and the corresponding transformation of the wave functions |ΨX⟩ and |ΨY⟩ such that the transition matrix elements of any operator may be calculated with very little additional computational effort compared to the case where |ΨX⟩ and |ΨY⟩ belong to the same MO basis. To this end, (21) the LU decomposition of the inverse of the orbital overlap matrix SXY (with SpqXY = ⟨ϕpX|ϕqY⟩) is constructed
(2)
The CXA and CYB matrices define the transformation from the MO to the biorthogonal basis such that
(3)
Before proceeding to the transformation of the wave functions |ΨX⟩, let us briefly introduce the wave function ansatz employed with DMRG, the MPS. A general CI ansatz for an arbitrary wave function |Ψ⟩ in a Hilbert space spanned by L spatial orbitals may be expressed as
(4)
with ck1...kL as the CI coefficients and |k1...kL⟩ as occupation number vectors. The notation |k1...kL⟩ reflects the fact that for each spatial orbital l, we may have a local basis state |kl⟩ = {|↑↓⟩, |↑⟩, |↓⟩, |0⟩} and the total occupation number vector consists of local occupations of all orbitals 1, ..., L.
The CI coefficients ck1...kL may be reshaped as an L-dimensional tensor and decomposed (38,39) by repeated application of the singular value decomposition into a product of matrices Mkl, yielding an MPS
(5)
The dimension of matrices (i. e., the a indices) may be limited to a certain maximum dimension m, usually referred to as the number of renormalized block states or maximum bond dimension. This way, the number of parameters entering the wave function ansatz definition is reduced from exponential, as it is in full CI, to polynomial. The optimization of MPS wave functions is most commonly carried out with the DMRG approach, for the explanation of which we refer the reader to the comprehensive reviews of Schollwöck (38,39) and ref (16).
Analogously to the MPS, operators may be expressed in a matrix product operator (MPO) form as
(6)
We consider next the transformation algorithm for wave functions |ΨX⟩, when |ΨX⟩ are MPSs, as introduced in ref (24). We perform another LU decomposition, this time of the CXA matrix, and from its lower and upper triangular parts (CLXA and CUXA, respectively), we construct the matrix t, with its lower and upper triangular part being
(7)
(8)
The matrix t is then used to transform the wave functions |ΨX⟩ as follows

  • First, the inactive orbitals are transformed by scaling the MPS with a factor α given by

    (9)
    where i runs over all inactive orbitals.

  • For the subsequent transformation with respect to the active orbitals, the following steps are repeated for each active orbital l:

    i

    Each matrix Mkl is multiplied with tll2 for kl = |↑↓⟩ and with tll for kl = |↑⟩ and |↓⟩,

    ii

    an MPO is applied to the scaled MPS |Ψ̃X⟩ yielding a transformed MPS

    (10)
    with
    (11)
    and
    (12)

    iii

    In the last step, one performs an SVD compression of |Ψ̃A⟩ to obtain the final MPS |ΨA⟩, a representation of the original state in the biorthonormal basis {ϕA}. Analogously, |ΨB⟩ may be constructed by repeating the steps mentioned above with the CYB matrix and |ΨY⟩ MPS. |ΨA⟩ and |ΨB⟩ are then employed to calculate transition density matrix elements and properties.

While the original MPS transformation algorithm has been shown to be highly accurate for various properties, including spin–orbit couplings and g-factors of actinides, (24) its current implementation has two major bottlenecks.
The first bottleneck originates from the SVD compression from step (3): the application of the MPO in eq 10 to an MPS results in a transformed MPS with the maximum bond dimension of b × m, where b is the maximum bond dimension of the MPO and m is the maximum bond dimension of the MPS |Ψ̃X⟩. The final SVD compression in step (3) reduces the final bond dimension of the transformed MPS, which is necessary since the storage size of the MPS and the cost of transition density matrix element evaluation (40) scales with and thus becomes prohibitively expensive for large m. However, MPS compression itself is a computationally expensive step with a computational cost of and constitutes a crucial bottleneck in MPSSI. The original MPSSI implementation (24) employs a fixed value of m = 8000, preserving the expectation value of the energy up to 10–8 a. u., but at a price of significant computational cost.
The second bottleneck arises from the construction of the MPO : as shown in eq 11, this step requires the construction of the operator, which is not trivial. The original implementation (24) avoids this problem by calculating and and adding the resulting MPS afterward. Applying the operator twice, however, increases the bond dimension of the resulting MPS by a factor of b2 and requires an additional MPS compression step between the first and the second application of .
In this work, we improve the efficiency of the MPSSI method by introducing two simple but effective changes to the MPS transformation algorithm. The first is the first-order approximation of eq 11 by neglecting the final second-order term. This approximation may be justified as follows: since to allow for the formation of biorthogonal bases, the original MO bases have to be sufficiently similar, i. e., show an overlap fairly close to unity, the resulting t matrix should not deviate significantly from the identity matrix. Equation 11 can be thought of as being a second-order Taylor approximation to the exponential of , and in the regime of t close to identity also, a linear approximation should hold. While accounting through the application of for a full rotation of a singly occupied orbital j in a given many-particle basis state of the complete many-particle wave function, neglecting the term corresponds to an approximation of the full effect of the rotation for a corresponding doubly occupied orbital k. In general, the latter requires the application of a two-electron excitation operator epkqk = EpkEqk – δkqEpk. (21) Hence, neglecting , as proposed in the present work, corresponds to approximating the two-electron excitation operator epkqk for the transformation of a doubly occupied orbital k in a given many-particle basis state by a sum of one-electron excitation operators, that is, epkqk ≈ Eqk – δkqEpk. Furthermore, eq 12 shows that the operator is scaled with the ratio between the off-diagonal and diagonal elements of . For an orthogonal basis, will be the identity matrix and this ratio will be zero. Therefore, a simple estimate based on off-diagonal elements of t, such as the L2 norm of t – IL (with IL as an L × L identity matrix), may be employed as a measure of the accuracy of the approximation.
The second approximation is the reduction of the maximum bond dimension of the compressed MPS, therefore reducing the computational cost of the MPS compression. We highlight that the speedup associated with the reduction of the maximum bond dimension employed in the MPS compression step comes at the price of losing accuracy in the approximation of the full-CI wave function as an MPS. Specifically, the error associated with this truncation step will increase with the difference between the two sets of nonorthogonal molecular orbitals, as will be demonstrated later in the results. However, two sets of molecular orbitals that are obtained for two different spin configurations and based on the same molecular structure are often not drastically different. This is the reason why, as we will show in the following, the MPS can be largely compressed without compromising the accuracy of the matrix elements of the spin–orbit coupling operator. The compression scheme may become less efficient in more complex cases, such as for calculating transition properties between orbitals obtained for different excited states calculated at different nuclear geometries. Still, it would be possible in these cases to adapt the bond dimension m to yield a given target wave function accuracy that is selected a priori, as discussed in ref (41). Alternatively, the loss of accuracy consequent to the MPS compression can be monitored by calculating the expectation value of operators that are associated with conserved quantum numbers before and after the truncation. As we showed in our original work on MPSSI, a small change in the squared spin operator indicates that the wave function accuracy is preserved after the truncation step.
In the following section, we demonstrate that both approaches significantly improve the computational cost of MPSSI with almost no effect on accuracy. Although both steps reduce the accuracy of the transformation, the following numerical test demonstrate that the errors introduced are negligible for several types of properties.

3. Numerical Examples

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As a testbed we employ trans, trans, trans-[Pt(N3)2(OH)2(NH3)2] (in the following referred to as 1), which is a flagship Pt(IV) azide complex, relevant in photoactivated cancer chemotherapy. (42−44) As the majority of 5d metal compounds, 1 shows strong spin–orbit couplings and since its photoactivation mechanism involves azide dissociation, such a process is best described by multiconfigurational methods. (45)

3.1. Performance of MPSSI Approximation on Wave Function Overlaps

In principle, CASSI/MPSSI allows for an easy calculation of wave function overlaps constructed with nonorthogonal orbital sets. Wave function overlaps, especially between states at different molecular structures or spin multiplicities, are widely used in ab initio excited-state molecular dynamics (19,36,46,47) or in wave function analysis. (48) Here, we investigated the accuracy of the linear approximation to (in the following called “MPSSI approximation”) for wave function overlaps of both ground and excited states for varying molecular structures of the same molecule. With an increasing deviation of molecular structures, the dissimilarity of the orbitals and the L2 norm of the t – IL matrix also increases, allowing us to also assess the limits of the MPSSI approximation with the increasing norm.
We performed CASSCF and DMRG-SCF calculations with a comparably small active space of eight electrons in nine orbitals. This active space is capable to qualitatively describe the energies of the lowest excited states and is also small enough for DMRG-SCF to be able to reproduce the CASSCF results almost exactly: the final DMRG-SCF energies differ from their CASSCF counterparts by no more than 10–7 a. u.
We performed a rigid scan along the Pt–N bond of one of the azide ligands with CASSCF and DMRG-SCF and calculated the wave function overlap of the lowest five singlet states at structures with an elongated Pt–N bond with their counterparts at the equilibrium structure. We calculated the overlaps of the CASSCF wave functions with CASSI and those of DMRG-SCF wave functions with full and approximate MPSSI: the average pairwise differences between these are shown in Figure 1a. The overlap difference between CASSCF and full MPSSI (green line) reflects the error arising only due to DMRG approximation to the CASSCF wave function. The effect arising due to the MPSSI approximation can be fully estimated from the approximate to full MPSSI difference (red line). The corresponding changes in the L2 norm of t – IL matrices are shown in Figure 1b.

Figure 1

Figure 1. (a) Average differences of overlaps calculated with CASSCF and full and approximate MPSSI; (b) L2 norm of t – IL matrices, with IL as the identity matrix; XA corresponds to the orbitals at the equilibrium structure and YB to orbitals at a given r – req.

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Given the tightly converged DMRG-SCF wave function, the errors arising due to the DMRG approximation are negligible: for all r – req values except 2.8 Å, the overlap error is less than 3 × 10–4, whereas for the latter calculation, it rises slightly to 2 × 10–3. This discrepancy is due to a slightly poorer convergence of the wave function at this particular r – req value than for other Pt–N bond lengths. Recalling that in contrast to the quadratic convergence of the energy, property calculations converge linearly with respect to the wave function quality, the maximum energy error for this case is closer to 10–7 a. u., whereas for other Pt–N bond lengths, it is well below this value. Nevertheless, all of these errors are so small that they may be considered negligible. The MPSSI approximation error is, however, larger than the DMRG approximation error for all calculations and rises with increasing t norm: starting with approximately 4 × 10–4 at r – req = 0.2 Å with a corresponding t – IL norm of 0.4 (and thus remaining in the same order of magnitude as the DMRG approximation errors), it steadily increases with increasing t – IL norm, reaching values of 8 × 10–3 for the extended Pt–N bonds.
In the range of r – req of 1.2 to 1.8 Å, we see a particularly large increase in the MPSSI approximation error, which corresponds to t – IL norm values between 0.9 and 1. Therefore, we propose a conservative cutoff t – IL norm value of 1, below which we recommend to use the approximation. This choice is, however, largely arbitrary: the average overlap error at the cutoff value is 2 × 10–3, and even the largest error value of 8 × 10–3 in these calculation series is still sufficient for a qualitatively correct calculation.
The suitability of larger MPSSI approximation errors for qualitative calculations is best illustrated if one compares the results to those from a partially converged DMRG-SCF calculation, which is a common practice in the literature. Figure 2 shows the same overlap errors displayed in Figure 1a but for partially converged DMRG-SCF wave functions, where energy differences to the corresponding CASSCF wave function are up to 2 × 10–4 a. u. The norms of the t – IL matrices are similar and the MPSSI approximation errors are almost the same as the corresponding errors for the fully converged DMRG-SCF wave functions. However, the errors arising due to the DMRG approximation increase sharply with the decreasing DMRG-SCF wave function quality. For energy errors in the range of 10–4 a. u. to 10–5 a. u., typical for large-scale DMRG calculations, the order of magnitude of the MPSSI approximation and the DMRG approximation error is similar, and therefore, approximate MPSSI is still suitable for qualitative calculations.

Figure 2

Figure 2. Average overlap error for overlaps with CASSCF and full and approximate MPSSI, for a partially converged DMRG-SCF wave function. Maximum energy error with respect to a corresponding CASSCF calculation is shown in black.

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We may conclude that the MPSSI approximation error is independent of the DMRG wave function quality but rather depends only on the t matrix. Thus, the L2 norm of the t – IL matrix constitutes an easy metric available prior to the MPSSI rotation that allows a simple decision whether the MPSSI approximation should be employed or not.

3.2. Performance of MPSSI Approximation on Spin–Orbit Couplings

Here, we investigate the MPSSI approximation performance in the calculation of spin–orbit coupling matrix elements, which is another typical use case for MPSSI. We employ the same active space of eight electrons in nine orbitals as in the previous example but calculate energies and spin–orbit coupling matrix elements for the five lowest singlets and triplet states of 1 at the equilibrium structure.
Figure 3a shows spin–orbit couplings calculated with CASSCF and DMRG-SCF employing the original (full) (24) and approximate MPSSI scheme. As in the previous example, DMRG-SCF wave functions have been converged so that the DMRG-SCF energies differ from their CASSCF counterparts by less than 10–7 a. u.: therefore, any error arising from the DMRG approximation is negligible. The differences between the calculated values are displayed in Figure 3b and show that the effect of the DMRG approximation (green curve) is indeed negligible: the largest error due to the DMRG approximation does not exceed 0.02 cm–1. The error due to the MPSSI approximation is, similarly to the previous example, slightly larger but still negligible for all practical purposes: the average error is 0.077 cm–1 and the maximum error is approximately 0.8 cm–1.

Figure 3

Figure 3. (a) Spin–orbit coupling (SOC) matrix elements for the first five singlet and five triplet states at the equilibrium structure of 1 for an active space of eight electrons in nine orbitals. (b) Absolute errors of the SOC matrix elements in panel (a).

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As in the previous example, we also consider the case of a partially converged DMRG-SCF wave function, where the energies of some states differ up to 10–5 a. u. from their CASSCF counterparts. This accuracy is typical for large-scale DMRG-SCF calculations and is more than sufficient for accurate absorption energies up to 10–5 a. u. The results are displayed in Figure 4. We note that in this case, the SOC error arising from the DMRG-SCF approximation increases by several orders of magnitude up to 30 cm–1, while the MPSSI approximation error remains the same. Thus, in this case, the total error in the DMRG calculation largely consists of the DMRG approximation error, while the MPSSI approximation error is completely negligible.

Figure 4

Figure 4. (a) Spin–orbit coupling (SOC) matrix elements for the first five singlet and five triplet states at the equilibrium structure of 1 with a partially converged DMRG-SCF wave function. (b) Absolute errors of the SOC matrix elements in panel (a).

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Figure 5 shows the absolute errors of spin–orbit corrected energies. All errors remain below 10–6 a. u. and thus negligible.

Figure 5

Figure 5. Errors of spin–orbit corrected energies calculated with CASSI and full and approximate MPSSI at the equilibrium structure of 1 for the first 20 spin–orbit coupled states arising from spin–orbit coupling of five singlets and five triplets. The active space employed in this calculation consisted of eight electrons in nine orbitals.

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Finally, we would like to mention the computational time savings arising from the approximation. Due to the small size of the active space, the MPSSI approximation is not the bottleneck in this calculation, but it already reduces the computational time by approximately 40%, that is, from 10 min 7 s to 6 min 6 s of run time on 4 cores of an Intel Xeon E5-2650 CPU.

3.3. Performance of the Methods with a Larger Active Space

From a calculation using time-dependent density functional theory (TD-DFT, CAM-B3LYP/def2-TZVPP) and including several low-lying singlet excited states of 1, we know that CASSCF and DMRG-SCF calculations with an active space of eight electrons in nine orbitals, as employed in the previous section, cannot even qualitatively account for the spin–orbit couplings: the largest absolute value for the spin–orbit coupling between the five lowest singlet and triplet excited states was 434 cm–1, whereas the corresponding value from a TD-DFT calculation was found to be approximately 1800 cm–1.
This insufficiency can be remedied by a DMRG-SCF calculation with 26 electrons in 19 orbitals, as employed in ref (45). As this active space is computationally too expensive for a CASSCF calculation, only DMRG-SCF calculations with subsequent approximate and full MPSSI calculations are performed. In addition, we assess the error arising due to the MPS compression step in the MPS transformation by testing various m values for the compressed MPS: the DMRG-SCF calculations were performed for m = 500, but during the MPSSI procedure, the intermediate MPS during rotation was compressed either to the original m = 500 or to m = 2000. Note that we could not afford a postcompression m value of 8000 from the original paper of Knecht et al. (24) due to its prohibitive computational requirements. Furthermore, in the following, we consider the 10 lowest singlet and 9 triplet states. The calculated SOC and their errors are shown in Figure 6.

Figure 6

Figure 6. (a) Spin–orbit coupling (SOC) matrix elements for the first 10 singlet and 9 triplet states at the equilibrium structure of 1, for an active space of 26 electrons in 19 orbitals. (b) Absolute errors of the SOC matrix elements in panel (a).

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As can be seen from Figure 6a, all methods yield almost identical values for the SOC. A closer look at errors (Figure 6b) reveals a maximum error of approximately 15 cm–1, which arises entirely due to the MPS compression. The MPSSI approximation error is negligible: the maximum MPSSI approximation error is 0.41 cm–1 for m = 500 and just 1 × 10–3 cm–1 for m = 2000. The small MPSSI approximation error is not surprising for this calculation, as the t – IL norms are only 0.002 and 0.006. We can also see that the MPSSI approximation is affected by compression but only very slightly: it is the compression error in the first place that contributes to the total error, which is nevertheless still small enough for quantitative results.
The errors in the spin–orbit corrected energies are analyzed in Figure 7. Also, here, the largest error in energies arises solely due to the compression. With an average error of 1.3 × 10–5 a. u. or 3 × 10–4 eV, it is also almost negligible. The MPSSI approximation error alone for both m = 500 and 2000 are at least 2 orders of magnitude smaller and are at the same order of magnitude as typical convergence thresholds for the SCF procedure and way lower than the expected DMRG truncation error: it can be safely neglected. It is noteworthy, however, that the MPSSI approximation error increases slightly for the lower-quality m = 500 DMRG wave function, implying a small direct effect of the compression on the approximation.

Figure 7

Figure 7. Errors of spin–orbit corrected energies calculated with full and approximate MPSSI for m = 500 and m = 2000 at the equilibrium structure of 1 for the first 37 spin–orbit coupled states. The active space employed in this calculation consisted of 26 electrons in 19 orbitals.

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Table 1 discloses the massive speedup that both the MPSSI approximation and the compression entail. Compared to a full MPSSI calculation with m = 2000, the MPSSI approximation gains a 7.5-fold speedup, the compression alone to m = 500 gains a 18-fold speedup, and the combination of both methods gains an overwhelming 63-fold speedup. Both, the MPSSI approximation and compression are essentially able to eliminate the bottlenecks in the MPSSI method, while still retaining for quantitative accuracy.
Table 1. Runtimes for Approximate and Full MPSSI Calculations with Intermediate and Final MPS Compression to m = 500 and 2000, as Run on 24 Cores of an AMD EPYC 7502 CPU
mapprox.Full
5006 h 2 m21 h 46 m
20002 d 3 h15 d 19 h
Although the smallest m value of 500 chosen by us is dictated by the original m value employed during the wave function optimization, it is tempting to use an even smaller value to save further computational time. However, given the comparably larger compression error that would increase even further for smaller m values, we do not recommend such a reduction.

4. Conclusions

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In this work, we presented two modifications to the original formulation of the MPSSI method by Knecht et al., (24) which despite being simple allow for drastic computational savings while retaining controlled accuracy in the DMRG-SCF calculation of properties.
The first modification, named the “MPSSI approximation”, is based on the omission of the quadratic term in the operator that is employed to counterrotate the MPS, to match the effect of the basis transformation. The second modification consists of decreasing the maximum bond dimension of the intermediate and the final counterrotated MPS by the SVD compression with a smaller m value. The accuracy of both modifications may be controlled independently of each other by a numerical parameter. In the case of the MPSSI approximation, it is the L2 norm of the t – IL matrix employed for the orbital rotation, which is known before the time-consuming MPS counterrotation, and thus allows for an error estimate of the MPSSI approximation beforehand. For the MPS compression, it is the m value of the intermediate and the final compressed MPS.
We have tested both modifications in two common useful scenarios where efficiency is highly desired: the calculation of wave function overlaps and spin–orbit couplings. Both quantities are, for example, indispensable to perform efficient ab initio nonadiabatic simulations on the fly. In all our examples, the discrepancies in these properties due to the MPSSI approximation error were very small. For tightly converged DMRG-SCF wave functions, close enough to CASSCF wave functions, the MPSSI approximation error was found to be larger than the DMRG approximation error but unlike the latter not dependent on the wave function quality. Instead, it shows monotonous dependence on the L2 norm of the t – IL matrix. When DMRG-SCF employs large active spaces, the MPS compression to the original m value of the unrotated MPS allows for very substantial computational time savings but introduces an additional source of error: although the MPS compression error is larger than that of the MPSSI approximation, it is still small enough to allow quantitative computation of properties.
In the current calculations, the MPSSI approximation and the MPS compression to m = 500 gave us a total 63-fold speedup compared to a calculation with compression to m = 2000, while maintaining a total error still small enough for quantitative computation of properties. The compression to m = 8000 as in the original implementation could not be performed due to excessive computational requirements: the performance gain compared to such a calculation would have been even larger. We believe that the speedups achieved with the improvements in this work will pave the way to faster and more affordable large-scale multiconfigurational calculations, as well as allow DMRG-SCF to be applied in computationally intensive scenarios, for example, in ab-initio excited-state molecular dynamic simulations.

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.1c00674.

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Acknowledgments

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L.F. thanks the Austrian Science Fund (FWF) for generous funding via the Schrödinger fellowship (project no. J 3935-N34). The University of Vienna is thanked for continuous support and the VSC for the computational resources.

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This article references 48 other publications.

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    A new large-scale parallel MC-SCF (MCSCF) implementation in the open-source NWChem computational chem. code is presented. The generalized active space approach is used to partition large CI vectors and generate a sufficient no. of batches that can be distributed to the available cores. Massively parallel CI calcns. with large active spaces can be performed. The new parallel MCSCF implementation is tested for the chromium trimer and for an active space of 20 electrons in 20 orbitals, which can now routinely be performed. Unprecedented CI calcns. with an active space of 22 electrons in 22 orbitals for the pentacene systems were performed and a single CI iteration calcn. with an active space of 24 electrons in 24 orbitals for the chromium tetramer was possible. The chromium tetramer corresponds to a CI expansion of one trillion Slater determinants (914 058 513 424) and is the largest conventional CI calcn. attempted up to date. (c) 2017 American Institute of Physics.
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    We have compared different strategies for ab initio quantum chem. d. matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the ref. complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approx. occupation nos. or a priori, using a Moller-Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping const. the no. m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate ests. of the full CI energy, making the expansion independent on the actual values of the two parameters m and λ. We propose to generate the factors for the two blocks from ground and excited eigenvectors of the Hamiltonian matrix.
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    A review. In the past two decades, the d. matrix renormalization group (DMRG) has emerged as an innovative new method in quantum chem. relying on a theor. framework very different from that of traditional electronic structure approaches. The development of the quantum chem. DMRG has been remarkably fast: it has already become one of the ref. approaches for large-scale multiconfigurational calcns. This perspective discusses the major features of DMRG, highlighting its strengths and weaknesses also in comparison with other novel approaches. The method is presented following its historical development, starting from its original formulation up to its most recent applications. Possible routes to recover dynamical correlation are discussed in detail. Emerging new fields of applications of DMRG are explored, such as its time-dependent formulation and the application to vibrational spectroscopy. (c) 2020 American Institute of Physics.
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    We present an approach that allows to produce the two-body d. matrix during the d. matrix renormalization group (DMRG) run without an addnl. increase in the current disk and memory requirements. The computational cost of producing the two-body d. matrix is proportional to O(M3k2 + M2k4). The method is based on the assumption that different elements of the two-body d. matrix can be calcd. during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theor. structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body d. matrix free from the N-representability problem. We explain the problem of local min. that may be encountered in the DMRG calcns. We discuss theor. why and when the one- and two-site DMRG procedures may get stuck in a metastable soln., and we list practical solns. helping the minimization to avoid the local min. (c) 2008 American Institute of Physics.
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    Transition probability calculations for atoms using nonorthogonal orbitals
    Olsen, Jeppe; Godefroid, Michel R.; Joensson, Per; Malmqvist, Per Ake; Fischer, Charlotte Froese
    Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics (1995), 52 (4-B), 4499-508CODEN: PLEEE8; ISSN:1063-651X. (American Physical Society)
    Individual orbital optimization of wave functions for the initial and final states produces the most accurate wave functions for given expansions, but complicates the calcn. of transition-matrix elements since the two sets of orbitals will be nonorthogonal. The orbital sets can be transformed to become biorthonormal, in which case the evaluation of any matrix element can proceed as in the orthonormal case. The transformation of the wave-function expansion to the new basis imposes certain requirements on the wave function, depending on the type of transformation. An efficient and general method was found a few years ago for expansions in determinants, spin-coupled configurations, or configuration state functions for mols. belonging to the D2h point group or its subgroups. The method requires only that the expansions are closed under deexcitation and thus applies to restricted active space wave functions. This type of expansion is efficient for correlation studies and includes many types of expansions as special cases. The above technique has been generalized to the at., symmetry adapted case requiring the treatment of degenerate shells nlN, with arbitrary occupation nos. 0 ≤ N ≤ 4l + 2. A computer implementation of the algorithm in the multiconfiguration Hartree-Fock at.-structure package for atoms allows the calcn. of transition moments for individually optimized states. An application is presented for the BI 1s22s12p2Po → 1s22s2p22D elec. dipole transition probability, which is highly sensitive to core-polarization effects.
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    Malmqvist, P. A.; Rendell, A.; Roos, B. O. The Restricted Active Space Self-Consistent-Field Method, Implemented with a Split Graph Unitary Group Approach. J. Phys. Chem. 1990, 94, 5477– 5482,  DOI: 10.1021/j100377a011
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    The restricted active space self-consistent-field method, implemented with a split graph unitary group approach
    Malmqvist, Per Aake; Rendell, Alistair; Roos, Bjoern O.
    Journal of Physical Chemistry (1990), 94 (14), 5477-82CODEN: JPCHAX; ISSN:0022-3654.
    An MCSCF method based on a restricted active space (RAS) type wave function has been implemented. The RAS concept is an extension of the complete active space (CAS) formalism, where the active orbitals are partitioned into three subspaces: RAS1, which contains up to a given max. no. of holes; RAS2, where all possible distributions of electrons are allowed; and RAS3, which contains up to a given max. no. of electrons. A typical example of a RAS wave function is all single, double, etc. excitations with a CAS ref. space. Spin-adapted configurations are used as the basis for the MC expansion. Vector coupling coeffs. are generated by using a split graph variant of the unitary group approach, with the result that very large MC expansions (up to around 106) can be treated, still keeping the no. of coeffs. explicitly computed small enough to be kept in central storage. The largest MCSCF calcn. that has so far been performed with the new program is for a CAS wave function contg. 716,418 CSF's. The MCSCF optimization is performed using an approx. form of the super-CI method. A quasi-Newton update method is used as a convergence accelerator and results in much improved convergence in most applications.
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    Knecht, S.; Keller, S.; Autschbach, J.; Reiher, M. A Nonorthogonal State-Interaction Approach for Matrix Product State Wave Functions. J. Chem. Theory Comput. 2016, 12, 5881– 5894,  DOI: 10.1021/acs.jctc.6b00889
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    A Nonorthogonal State-Interaction Approach for Matrix Product State Wave Functions
    Knecht, Stefan; Keller, Sebastian; Autschbach, Jochen; Reiher, Markus
    Journal of Chemical Theory and Computation (2016), 12 (12), 5881-5894CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present a state-interaction approach for matrix product state (MPS) wave functions in a nonorthogonal MO basis. Our approach allows us to calc. for example transition and spin-orbit coupling matrix elements between arbitrary electronic states provided that they share the same one-electron basis functions and active orbital space, resp. The key element is the transformation of the MPS wave functions of different states from a nonorthogonal to a biorthonormal MO basis representation exploiting a sequence of non-unitary transformations following a proposal by Malmqvist (Int. J. Quantum Chem.30, 479 (1986)). This is well-known for traditional wave-function parametrizations but has not yet been exploited for MPS wave functions.
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    Roemelt, M. Spin Orbit Coupling for Molecular Ab Initio Density Matrix Renormalization Group Calculations: Application to g-Tensors. J. Chem. Phys. 2015, 143, 044112,  DOI: 10.1063/1.4927432
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    Spin orbit coupling for molecular ab initio density matrix renormalization group calculations: Application to g-tensors
    Roemelt, Michael
    Journal of Chemical Physics (2015), 143 (4), 044112/1-044112/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Spin Orbit Coupling (SOC) is introduced to mol. ab initio d. matrix renormalization group (DMRG) calcns. In the presented scheme, one 1st approximates the electronic ground state and a no. of excited states of the Born-Oppenheimer (BO) Hamiltonian with the aid of the DMRG algorithm. Owing to the spin-adaptation of the algorithm, the total spin S is a good quantum no. for these states. After the nonrelativistic DMRG calcn. is finished, all magnetic sublevels of the calcd. states are constructed explicitly, and the SOC operator is expanded in the resulting basis. To this end, spin orbit coupled energies and wavefunctions were obtained as eigenvalues and eigenfunctions of the full Hamiltonian matrix which is composed of the SOC operator matrix and the BO Hamiltonian matrix. This treatment corresponds to a quasi-degenerate perturbation theory approach and can be regarded as the mol. equiv. to at. Russell-Saunders coupling. For the evaluation of SOC matrix elements, the full Breit-Pauli SOC Hamiltonian is approximated by the widely used spin-orbit mean field operator. This operator allows for an efficient use of the 2nd quantized triplet replacement operators that are readily generated during the nonrelativistic DMRG algorithm, together with the Wigner-Eckart theorem. With a set of spin-orbit coupled wavefunctions at hand, the mol. g-tensors are calcd. following the scheme proposed by Gerloch and McMeeking. It interprets the effective mol. g-values as the slope of the energy difference between the lowest Kramers pair with respect to the strength of the applied magnetic field. Test calcns. on a chem. relevant Mo complex demonstrate the capabilities of the presented method. (c) 2015 American Institute of Physics.
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    Sayfutyarova, E. R.; Chan, G. K.-L. A State Interaction Spin-Orbit Coupling Density Matrix Renormalization Group Method. J. Chem. Phys. 2016, 144, 234301,  DOI: 10.1063/1.4953445
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    A state interaction spin-orbit coupling density matrix renormalization group method
    Sayfutyarova, Elvira R.; Chan, Garnet Kin-Lic
    Journal of Chemical Physics (2016), 144 (23), 234301/1-234301/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The authors describe a state interaction spin-orbit (SISO) coupling method using d. matrix renormalization group (DMRG) wavefunctions and the spin-orbit mean-field (SOMF) operator. The authors implement the authors' DMRG-SISO scheme using a spin-adapted algorithm that computes transition d. matrixes between arbitrary matrix product states. To demonstrate the potential of the DMRG-SISO scheme the authors present accurate benchmark calcns. for the zero-field splitting of the copper and gold atoms, comparing to earlier complete active space SCF and 2nd-order complete active space perturbation theory results in the same basis. The authors also compute the effects of spin-orbit coupling on the spin-ladder of the iron-sulfur dimer complex [Fe2S2(SCH3)4]3-, detg. the splitting of the lowest quartet and sextet states. The magnitude of the zero-field splitting for the higher quartet and sextet states approaches a significant fraction of the Heisenberg exchange parameter. (c) 2016 American Institute of Physics.
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    Sayfutyarova, E. R.; Chan, G. K.-L. Electron Paramagnetic Resonance G-Tensors from State Interaction Spin-Orbit Coupling Density Matrix Renormalization Group. J. Chem. Phys. 2018, 148, 184103,  DOI: 10.1063/1.5020079
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    Electron paramagnetic resonance g-tensors from state interaction spin-orbit coupling density matrix renormalization group
    Sayfutyarova, Elvira R.; Chan, Garnet Kin-Lic
    Journal of Chemical Physics (2018), 148 (18), 184103/1-184103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We present a state interaction spin-orbit coupling method to calc. ESR g-tensors from d. matrix renormalization group wavefunctions. We apply the technique to compute g-tensors for the TiF3 and CuCl2-4 complexes, a [2Fe-2S] model of the active center of ferredoxins, and a Mn4CaO5 model of the S2 state of the oxygen evolving complex. These calcns. raise the prospects of detg. g-tensors in multireference calcns. with a large no. of open shells. (c) 2018 American Institute of Physics.
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    Knecht, S.; Legeza, Ö.; Reiher, M. Communication: Four-Component Density Matrix Renormalization Group. J. Chem. Phys. 2014, 140, 041101,  DOI: 10.1063/1.4862495
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    Communication. Four-component density matrix renormalization group
    Knecht, Stefan; Legeza, Ors; Reiher, Markus
    Journal of Chemical Physics (2014), 140 (4), 041101/1-041101/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We present the 1st implementation of the relativistic quantum chem. 2- and 4-component d. matrix renormalization group algorithm that includes a variational description of scalar-relativistic effects and spin-orbit coupling. Numerical results based on the 4-component Dirac-Coulomb Hamiltonian are presented for the std. ref. mol. for correlated relativistic benchmarks: TlH. (c) 2014 American Institute of Physics.
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    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.7b01065
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    Efficient Relativistic Density-Matrix Renormalization Group Implementation in a Matrix-Product Formulation
    Battaglia, Stefano; Keller, Sebastian; Knecht, Stefan
    Journal 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.
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    Crespo-Otero, R.; Barbatti, M. Recent Advances and Perspectives on Nonadiabatic Mixed Quantum-Classical Dynamics. Chem. Rev. 2018, 118, 7026– 7068,  DOI: 10.1021/acs.chemrev.7b00577
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    Recent Advances and Perspectives on Nonadiabatic Mixed Quantum-Classical Dynamics
    Crespo-Otero, Rachel; Barbatti, Mario
    Chemical Reviews (Washington, DC, United States) (2018), 118 (15), 7026-7068CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
    A review. Nonadiabatic mixed quantum-classical (NA-MQC) dynamics methods form a class of computational theor. approaches in quantum chem. tailored to investigate the time evolution of nonadiabatic phenomena in mols. and supramol. assemblies. NA-MQC is characterized by a partition of the mol. system into two subsystems: one to be treated quantum mech. (usually but not restricted to electrons) and another to be dealt with classically (nuclei). The two subsystems are connected through nonadiabatic couplings terms to enforce self-consistency. A local approxn. underlies the classical subsystem, implying that direct dynamics can be simulated, without needing precomputed potential energy surfaces. The NA-MQC split allows reducing computational costs, enabling the treatment of realistic mol. systems in diverse fields. Starting from the three most well-established methods - mean-field Ehrenfest, trajectory surface hopping, and multiple spawning - this review focuses on the NA-MQC dynamics methods and programs developed in the last 10 years. It stresses the relations between approaches and their domains of application. The electronic structure methods most commonly used together with NA-MQC dynamics are reviewed as well. The accuracy and precision of NA-MQC simulations are critically discussed, and general guidelines to choose an adequate method for each application are delivered.
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    Lengsfield, B. H., III; Saxe, P.; Yarkony, D. R. On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wave functions and analytic gradient methods. I. J. Chem. Phys. 1984, 81, 4549– 4553,  DOI: 10.1063/1.447428
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    On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wave functions and analytic gradient methods. I
    Lengsfield, Byron H., III; Saxe, Paul; Yarkony, David R.
    Journal of Chemical Physics (1984), 81 (10), 4549-53CODEN: JCPSA6; ISSN:0021-9606.
    A method for the efficient evaluation of nonadiabatic coupling matrix elements of the form 〈ΨI|.vdelta./.vdelta.RαΨJ〉 is presented. The wave function ΨI and ΨJ are assumed to be MCSCF wave functions optimized within the state averaged (SA) approxn. The method, which can treat several states simultaneously, derives its efficiency from the direct soln. of the coupled perturbed state averaged MCSCF equations and the availability of other appropriate deriv. integrals. An extension of this approach to SA-MCSCF/CI wave functions is described. Computational efficiencies can be achieved by exploiting analogies with analytic CI gradient methods. Numerical examples for C2v approach of Mg to H2 are presented.
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    Stålring, J.; Bernhardsson, A.; Lindh, R. Analytical Gradients of a State Average MCSCF State and a State Average Diagnostic. Mol. Phys. 2001, 99, 103– 114,  DOI: 10.1080/002689700110005642
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    Analytical gradients of a state average MCSCF state and a state average diagnostic
    Stalring, Jonna; Bernhardsson, Anders; Lindh, Roland
    Molecular Physics (2001), 99 (2), 103-114CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
    An efficient method for calcg. the Lagrange multipliers and the anal. gradients of one state included in a state av. MCSCF wave function is presented. The state av. energy of an "equal-wt." scheme is invariant to rotations within the state av. subspace and that the corresponding rotations should be eliminated from the Lagrangian equations. A diagnostic is presented, which gauges the energy difference between a state defined by a state av. calcn. and the corresponding fully variational multi-configurational SCF state.
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    Snyder, J. W.; Hohenstein, E. G.; Luehr, N.; Martínez, T. J. An Atomic Orbital-Based Formulation of Analytical Gradients and Nonadiabatic Coupling Vector Elements for the State-Averaged Complete Active Space Self-Consistent Field Method on Graphical Processing Units. J. Chem. Phys. 2015, 143, 154107,  DOI: 10.1063/1.4932613
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    An atomic orbital-based formulation of analytical gradients and nonadiabatic coupling vector elements for the state-averaged complete active space self-consistent field method on graphical processing units
    Snyder, James W.; Hohenstein, Edward G.; Luehr, Nathan; Martinez, Todd J.
    Journal of Chemical Physics (2015), 143 (15), 154107/1-154107/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We recently presented an algorithm for state-averaged complete active space SCF (SA-CASSCF) orbital optimization that capitalizes on sparsity in the AO basis set to reduce the scaling of computational effort with respect to mol. size. Here, we extend those algorithms to calc. the analytic gradient and nonadiabatic coupling vectors for SA-CASSCF. Combining the low computational scaling with acceleration from graphical processing units allows us to perform SA-CASSCF geometry optimizations for mols. with more than 1000 atoms. The new approach will make minimal energy conical intersection searches and nonadiabatic dynamics routine for mol. systems with O(102) atoms. (c) 2015 American Institute of Physics.
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    Snyder, J. W.; Fales, B. S.; Hohenstein, E. G.; Levine, B. G.; Martínez, T. J. A Direct-Compatible Formulation of the Coupled Perturbed Complete Active Space Self-Consistent Field Equations on Graphical Processing Units. J. Chem. Phys. 2017, 146, 174113,  DOI: 10.1063/1.4979844
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    A direct-compatible formulation of the coupled perturbed complete active space self-consistent field equations on graphical processing units
    Snyder, James W.; Fales, B. Scott; Hohenstein, Edward G.; Levine, Benjamin G.; Martinez, Todd J.
    Journal of Chemical Physics (2017), 146 (17), 174113/1-174113/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We recently developed an algorithm to compute response properties for the state-averaged complete active space SCF method (SA-CASSCF) that capitalized on sparsity in the AO basis. Our original algorithm was limited to treating small to moderate sized active spaces, but the recent development of graphical processing unit (GPU) based direct-CI algorithms provides an opportunity to extend this to large active spaces. We present here a direct-compatible version of the coupled perturbed equations, enabling us to compute response properties for systems treated with arbitrary active spaces (subject to available memory and computation time). This work demonstrates that the computationally demanding portions of the SA-CASSCF method can be formulated in terms of seven fundamental operations, including Coulomb and exchange matrix builds and their derivs., as well as, generalized one- and two-particle d. matrix and σ vector constructions. As in our previous work, this algorithm exhibits low computational scaling and is accelerated by the use of GPUs, making possible optimizations and nonadiabatic dynamics on systems with O(1000) basis functions and O(100) atoms, resp. (c) 2017 American Institute of Physics.
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    Mai, S.; Marquetand, P.; González, L. Nonadiabatic Dynamics: The SHARC Approach. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018, 8, e1370  DOI: 10.1002/wcms.1370
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    Granucci, G.; Persico, M.; Toniolo, A. Direct Semiclassical Simulation of Photochemical Processes with Semiempirical Wave Functions. J. Chem. Phys. 2001, 114, 10608– 10615,  DOI: 10.1063/1.1376633
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    Direct semiclassical simulation of photochemical processes with semiempirical wave functions
    Granucci, G.; Persico, M.; Toniolo, A.
    Journal of Chemical Physics (2001), 114 (24), 10608-10615CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The authors describe a new method for the simulation of excited state dynamics, based on classical trajectories and surface hopping, with direct semiempirical calcn. of the electronic wave functions and potential energy surfaces (DTSH method). Semiempirical self-consistent-field MOs (SCF MO's) are computed with geometry-dependent occupation nos., in order to ensure correct homolytic dissocn., fragment orbital degeneracy, and partial optimization of the lowest virtuals. Electronic wave functions are of the MO active space CI type, for which analytic energy gradients have been implemented. The time-dependent electronic wave function is propagated by means of a local diabatization algorithm which is inherently stable also in the case of surface crossings. The method is tested for the problem of excited ethylene nonadiabatic dynamics, and the results are compared with recent quantum mech. calcns.
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    Freitag, L.; Ma, Y.; Baiardi, A.; Knecht, S.; Reiher, M. Approximate Analytical Gradients and Nonadiabatic Couplings for the State-Average Density Matrix Renormalization Group Self-Consistent-Field Method. J. Chem. Theory Comput. 2019, 15, 6724– 6737,  DOI: 10.1021/acs.jctc.9b00969
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    37
    Approximate Analytical Gradients and Nonadiabatic Couplings for the State-Average Density Matrix Renormalization Group Self-Consistent-Field Method
    Freitag, Leon; Ma, Yingjin; Baiardi, Alberto; Knecht, Stefan; Reiher, Markus
    Journal of Chemical Theory and Computation (2019), 15 (12), 6724-6737CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present an approx. scheme for anal. gradients and nonadiabatic couplings for a state-av. d. matrix renormalization group self-consistent-field (SA-DMRG-SCF) wavefunction. Our formalism follows closely the state-av. complete active space self-consistent-field (SA-CASSCF) ansatz, which employs a Lagrangian, and the corresponding Lagrange multipliers are obtained from a soln. of the coupled-perturbed CASSCF (CP-CASSCF) equations. We introduce a definition of the MPS Lagrange multipliers based on a single-site tensor in a mixed-canonical form of the MPS, such that the sweep procedure is avoided in the soln. of the CP-CASSCF equations. We employ our implementation for the optimization of a conical intersection in 1,2-dioxetanone, where we are able to fully reproduce the SA-CASSCF result up to arbitrary accuracy.
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    Schollwöck, U. The Density-Matrix Renormalization Group in the Age of Matrix Product States. Ann. Phys. 2011, 326, 96– 192,  DOI: 10.1016/j.aop.2010.09.012
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    Schollwöck, U. The Density-Matrix Renormalization Group: A Short Introduction. Philos. Trans. R. Soc., A 2011, 369, 2643– 2661,  DOI: 10.1098/rsta.2010.0382
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    Keller, S.; Dolfi, M.; Troyer, M.; Reiher, M. An Efficient Matrix Product Operator Representation of the Quantum Chemical Hamiltonian. J. Chem. Phys. 2015, 143, 244118,  DOI: 10.1063/1.4939000
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    An efficient matrix product operator representation of the quantum chemical Hamiltonian
    Keller, Sebastian; Dolfi, Michele; Troyer, Matthias; Reiher, Markus
    Journal of Chemical Physics (2015), 143 (24), 244118/1-244118/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We describe how to efficiently construct the quantum chem. Hamiltonian operator in matrix product form. We present its implementation as a d. matrix renormalization group (DMRG) algorithm for quantum chem. applications. Existing implementations of DMRG for quantum chem. are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the addnl. flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - Abelian and non-Abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program. (c) 2015 American Institute of Physics.
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    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: Condens. Matter Mater. Phys. 2003, 67, 125114,  DOI: 10.1103/physrevb.67.125114
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    Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach
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    Dalton Transactions (2011), 40 (1), 262-268CODEN: DTARAF; ISSN:1477-9226. (Royal Society of Chemistry)
    The photodecompn. of the anticancer complex trans,trans,trans-[Pt(N3)2(OH)2(NH3)2] in acidic aq. soln., as well as in phosphate-buffered saline (PBS), induced by UVA light (centered at λ = 365 nm) has been studied by multinuclear NMR spectroscopy. The authors show that the photoreaction pathway in PBS, which involves azide release, differs from that in acidic aq. conditions, under which N2 is a major product. In both cases, a no. of trans-{N-Pt(II/IV)-NH3} species were also obsd. as photoproducts, as well as the evolution of O2 and release of free ammonia with a subsequent increase in pH. The results from this study illustrate that photoinduced reactions of Pt(IV)-diazido derivs. can lead to novel reaction pathways, and therefore potentially to new cytotoxic mechanisms in cancer cells.
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    A review. Various PtIV diazides have been investigated over the years as light-activatable prodrugs that interfere with cell proliferation, accumulate in cancer cells and cause cell death. The potencies of the complexes vary depending on the substituted amines (pyridine = piperidine > ammine) as well as the coordination geometry (trans diazide > cis). Light-activated PtIV diazides tend to be less specific than cisplatin at inhibiting cancer cell growth, but cells resistant to cisplatin show little cross-resistance to PtIV diazides. Platinum is accumulated in the cancer cells to a similar level as cisplatin, but only when activated by light, indicating that reactive Pt species form photolytically. Studies show that Pt also becomes attached to cellular DNA upon the light activation of various PtIV diazides. Structures of some of the photolysis products were elucidated by LC-MS/MS; monoaqua- and diaqua-PtII complexes form that are reactive towards biomols. such as calf thymus DNA. Platination of calf thymus DNA can be blocked by the addn. of nucleophiles such as glutathione and chloride, further evidence that aqua-PtII species form upon irradn. Evidence is presented that reactive oxygen species may be generated in the first hours following photoactivation. Cell death does not take the usual apoptotic pathways seen with cisplatin, but appears to involve autophagy. Thus, photoactivated diazido-PtIV complexes represent an interesting class of potential anti-cancer agents that can be selectively activated by light and kill cells by a mechanism different to the anti-cancer drug cisplatin.
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    The Role of Triplet States in the Photodissociation of a Platinum Azide Complex by a Density Matrix Renormalization Group Method
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    Platinum azide complexes are appealing anticancer photochemotherapy drug candidates because they release cytotoxic azide radicals upon light irradn. Here we present a d. matrix renormalization group SCF (DMRG-SCF) study of the azide photodissocn. mechanism of trans,trans,trans-[Pt(N3)2(OH)2(NH3)2], including spin-orbit coupling. We find a complex interplay of singlet and triplet electronic excited states that falls into three different dissocn. channels at well-sepd. energies. These channels can be accessed either via direct excitation into barrierless dissociative states or via intermediate doorway states from which the system undergoes non-radiative internal conversion and intersystem crossing. The high d. of states, particularly of spin-mixed states, is key to aid non-radiative population transfer and enhance photodissocn. along the lowest electronic excited states.
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    We apply mol. dynamics with quantum transitions (MDQT), a surface-hopping method previously used only for electronic transitions, to proton transfer in soln., where the quantum particle is an atom. We use full classical mech. mol. dynamics for the heavy atom degrees of freedom, including the solvent mols., and treat the hydrogen motion quantum mech. We identify new obstacles that arise in this application of MDQT and present methods for overcoming them. We implement these new methods to demonstrate that application of MDQT to proton transfer in soln. is computationally feasible and appears capable of accurately incorporating quantum mech. phenomena such as tunneling and isotope effects. As an initial application of the method, we employ a model used previously by Azzouz and Borgis (1993) to represent the proton transfer reaction AH-B .dblharw. A--H+B in liq. Me chloride, where the AH-B complex corresponds to a typical phenol-amine complex. We have chosen this model, in part, because it exhibits both adiabatic and diabatic behavior, thereby offering a stringent test of the theory. MDQT proves capable of treating both limits, as well as the intermediate regime. Up to four quantum states were included in this simulation, and the method can easily be extended to include addnl. excited states, so it can be applied to a wide range of processes, such as photoassisted tunneling. In addn., this method is not perturbative, so trajectories can be continued after the barrier is crossed to follow the subsequent dynamics.
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    A simple and powerful method for comparing many-electron wavefunctions constructed at different levels of theory is presented. By using wavefunction overlaps, it is possible to analyze the effects of varying wavefunction models, MOs, and one-electron basis sets. The computation of wavefunction overlaps eliminates the inherent ambiguity connected to more rudimentary wavefunction anal. protocols, such as visualization of orbitals or comparing selected phys. observables. Instead, wavefunction overlaps allow processing the many-electron wavefunctions in their full inherent complexity. The presented method is particularly effective for excited state calcns. as it allows for automatic monitoring of changes in the ordering of the excited states. A numerical demonstration based on multireference computations of two test systems, the selenoacrolein mol. and an iridium complex, is presented. (c) 2016 American Institute of Physics.
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Cited By

This article is cited by 4 publications.

  1. Yang Guo, Ning Zhang, Wenjian Liu. SOiCISCF: Combining SOiCI and iCISCF for Variational Treatment of Spin–Orbit Coupling. Journal of Chemical Theory and Computation 2023, Article ASAP.
  2. Wenjian Liu. Perspective: Simultaneous treatment of relativity, correlation, and QED. WIREs Computational Molecular Science 2022, 88 https://doi.org/10.1002/wcms.1652
  3. Ning Zhang, Yunlong Xiao, Wenjian Liu. SOiCI and iCISO: combining iterative configuration interaction with spin–orbit coupling in two ways. Journal of Physics: Condensed Matter 2022, 34 (22) , 224007. https://doi.org/10.1088/1361-648X/ac5db4
  4. Haibo Ma, Ulrich Schollwöck, Zhigang Shuai. Density matrix renormalization group with orbital optimization. 2022, 149-188. https://doi.org/10.1016/B978-0-323-85694-2.00008-5
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  • Abstract

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    Figure 1

    Figure 1. (a) Average differences of overlaps calculated with CASSCF and full and approximate MPSSI; (b) L2 norm of t – IL matrices, with IL as the identity matrix; XA corresponds to the orbitals at the equilibrium structure and YB to orbitals at a given r – req.

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    Figure 2

    Figure 2. Average overlap error for overlaps with CASSCF and full and approximate MPSSI, for a partially converged DMRG-SCF wave function. Maximum energy error with respect to a corresponding CASSCF calculation is shown in black.

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    Figure 3

    Figure 3. (a) Spin–orbit coupling (SOC) matrix elements for the first five singlet and five triplet states at the equilibrium structure of 1 for an active space of eight electrons in nine orbitals. (b) Absolute errors of the SOC matrix elements in panel (a).

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    Figure 4

    Figure 4. (a) Spin–orbit coupling (SOC) matrix elements for the first five singlet and five triplet states at the equilibrium structure of 1 with a partially converged DMRG-SCF wave function. (b) Absolute errors of the SOC matrix elements in panel (a).

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    Figure 5

    Figure 5. Errors of spin–orbit corrected energies calculated with CASSI and full and approximate MPSSI at the equilibrium structure of 1 for the first 20 spin–orbit coupled states arising from spin–orbit coupling of five singlets and five triplets. The active space employed in this calculation consisted of eight electrons in nine orbitals.

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    Figure 6

    Figure 6. (a) Spin–orbit coupling (SOC) matrix elements for the first 10 singlet and 9 triplet states at the equilibrium structure of 1, for an active space of 26 electrons in 19 orbitals. (b) Absolute errors of the SOC matrix elements in panel (a).

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    Figure 7

    Figure 7. Errors of spin–orbit corrected energies calculated with full and approximate MPSSI for m = 500 and m = 2000 at the equilibrium structure of 1 for the first 37 spin–orbit coupled states. The active space employed in this calculation consisted of 26 electrons in 19 orbitals.

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  • References

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      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.
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    9. 9
      Mitrushenkov, A. O.; Fano, G.; Ortolani, F.; Linguerri, R.; Palmieri, P. Quantum Chemistry Using the Density Matrix Renormalization Group. J. Chem. Phys. 2001, 115, 6815– 6821,  DOI: 10.1063/1.1389475
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      Quantum chemistry using the density matrix renormalization group
      Mitrushenkov, Alexander O.; Fano, Guido; Ortolani, Fabio; Linguerri, Roberto; Palmieri, Paolo
      Journal of Chemical Physics (2001), 115 (15), 6815-6821CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A new implementation of the d. matrix renormalization group is presented for ab initio quantum chem. Test computations have been performed of the dissocn. energies of the diatomics Be2, N2, HF. A preliminary calcn. on the Cr2 mol. provides a new variational upper bound to the ground state energy.
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      Mitrushenkov, A. O.; Linguerri, R.; Palmieri, P.; Fano, G. Quantum Chemistry Using the Density Matrix Renormalization Group II. J. Chem. Phys. 2003, 119, 4148– 4158,  DOI: 10.1063/1.1593627
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      Quantum chemistry using the density matrix renormalization group II
      Mitrushenkov, A. O.; Linguerri, Roberto; Palmieri, Paolo; Fano, Guido
      Journal of Chemical Physics (2003), 119 (8), 4148-4158CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We have compared different strategies for ab initio quantum chem. d. matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the ref. complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approx. occupation nos. or a priori, using a Moller-Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping const. the no. m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate ests. of the full CI energy, making the expansion independent on the actual values of the two parameters m and λ. We propose to generate the factors for the two blocks from ground and excited eigenvectors of the Hamiltonian matrix.
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      Mitrushchenkov, A. O.; Fano, G.; Linguerri, R.; Palmieri, P. On the Importance of Orbital Localization in QC-DMRG Calculations. Int. J. Quantum Chem. 2011, 112, 1606– 1619,  DOI: 10.1002/qua.23173
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      Chan, G. K.-L.; Zgid, D. Chapter 7 The Density Matrix Renormalization Group in Quantum Chemistry. In Annual Reports in Computational Chemistry; Wheeler, R. A., Ed.; Elsevier, 2009; Vol. 5, pp 149– 162.
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      Marti, K. H.; Reiher, M. The Density Matrix Renormalization Group Algorithm in Quantum Chemistry. Z. Phys. Chem. 2010, 224, 583– 599,  DOI: 10.1524/zpch.2010.6125
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      The density matrix renormalization group algorithm in quantum chemistry
      Marti, Konrad Heinrich; Reiher, Markus
      Zeitschrift fuer Physikalische Chemie (Muenchen, Germany) (2010), 224 (3-4), 583-599CODEN: ZPCFAX; ISSN:0942-9352. (Oldenbourg Wissenschaftsverlag GmbH)
      A review. In this work, we derive the d. matrix renormalization group (DMRG) algorithm in the language of CI. Furthermore, the development of DMRG in quantum chem. is reviewed and DMRG-specific peculiarities are discussed. Finally, we discuss new results for a dinuclear μ-oxo bridged copper cluster, which is an important active-site structure in transition-metal chem., an area in which we pioneered the application of DMRG.
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      Wouters, S.; Van Neck, D. The Density Matrix Renormalization Group for Ab Initio Quantum Chemistry. Eur. Phys. J. D 2014, 68, 272,  DOI: 10.1140/epjd/e2014-50500-1
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      Baiardi, A.; Reiher, M. The Density Matrix Renormalization Group in Chemistry and Molecular Physics: Recent Developments and New Challenges. J. Chem. Phys. 2020, 152, 040903,  DOI: 10.1063/1.5129672
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      The density matrix renormalization group in chemistry and molecular physics: Recent developments and new challenges
      Baiardi, Alberto; Reiher, Markus
      Journal of Chemical Physics (2020), 152 (4), 040903CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A review. In the past two decades, the d. matrix renormalization group (DMRG) has emerged as an innovative new method in quantum chem. relying on a theor. framework very different from that of traditional electronic structure approaches. The development of the quantum chem. DMRG has been remarkably fast: it has already become one of the ref. approaches for large-scale multiconfigurational calcns. This perspective discusses the major features of DMRG, highlighting its strengths and weaknesses also in comparison with other novel approaches. The method is presented following its historical development, starting from its original formulation up to its most recent applications. Possible routes to recover dynamical correlation are discussed in detail. Emerging new fields of applications of DMRG are explored, such as its time-dependent formulation and the application to vibrational spectroscopy. (c) 2020 American Institute of Physics.
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      Freitag, L.; Reiher, M.. Quantum Chemistry and Dynamics of Excited States: Methods and Applications; González, L., Lindh, R., Eds.; Wiley, 2020; Vol. 1, pp 207– 246.
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      Zgid, D.; Nooijen, M. The Density Matrix Renormalization Group Self-Consistent Field Method: Orbital Optimization with the Density Matrix Renormalization Group Method in the Active Space. J. Chem. Phys. 2008, 128, 144116,  DOI: 10.1063/1.2883981
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      The density matrix renormalization group self-consistent field method: Orbital optimization with the density matrix renormalization group method in the active space
      Zgid, Dominika; Nooijen, Marcel
      Journal of Chemical Physics (2008), 128 (14), 144116/1-144116/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present the d. matrix renormalization group SCF (DMRG-SCF) approach that is analogous to the complete active space self-consisted field (CASSCF) method but instead of using for the description of the active space the full CI (FCI) method, the DMRG-SCF uses the d. matrix renormalization group (DMRG) method. The DMRG-SCF approach, similarly to CASSCF, properly describes the multiconfigurational character of the wave function but avoids the exponential scaling of the FCI method and replaces it with a polynomial scaling. Hence, calcns. for a larger no. of orbitals and electrons in the active space are possible since the DMRG method provides an efficient tool to automatically select from the full Hilbert space the many-body contracted basis states that are the most important for the description of the wave function. (c) 2008 American Institute of Physics.
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      Zgid, D.; Nooijen, M. Obtaining the Two-Body Density Matrix in the Density Matrix Renormalization Group Method. J. Chem. Phys. 2008, 128, 144115,  DOI: 10.1063/1.2883980
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      Obtaining the two-body density matrix in the density matrix renormalization group method
      Zgid, Dominika; Nooijen, Marcel
      Journal of Chemical Physics (2008), 128 (14), 144115/1-144115/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present an approach that allows to produce the two-body d. matrix during the d. matrix renormalization group (DMRG) run without an addnl. increase in the current disk and memory requirements. The computational cost of producing the two-body d. matrix is proportional to O(M3k2 + M2k4). The method is based on the assumption that different elements of the two-body d. matrix can be calcd. during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theor. structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body d. matrix free from the N-representability problem. We explain the problem of local min. that may be encountered in the DMRG calcns. We discuss theor. why and when the one- and two-site DMRG procedures may get stuck in a metastable soln., and we list practical solns. helping the minimization to avoid the local min. (c) 2008 American Institute of Physics.
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      Plasser, F.; Ruckenbauer, M.; Mai, S.; Oppel, M.; Marquetand, P.; González, L. Efficient and Flexible Computation of Many-Electron Wave Function Overlaps. J. Chem. Theory Comput. 2016, 12, 1207– 1219,  DOI: 10.1021/acs.jctc.5b01148
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      Efficient and Flexible Computation of Many-Electron Wave Function Overlaps
      Plasser, Felix; Ruckenbauer, Matthias; Mai, Sebastian; Oppel, Markus; Marquetand, Philipp; Gonzalez, Leticia
      Journal of Chemical Theory and Computation (2016), 12 (3), 1207-1219CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      A new algorithm for the computation of the overlap between many-electron wave functions is described. This algorithm allows for the extensive use of recurring intermediates and thus provides high computational efficiency. Because of the general formalism employed, overlaps can be computed for varying wave function types, MOs, basis sets, and mol. geometries. This paves the way for efficiently computing nonadiabatic interaction terms for dynamics simulations. In addn., other application areas can be envisaged, such as the comparison of wave functions constructed at different levels of theory. Aside from explaining the algorithm and evaluating the performance, a detailed anal. of the numerical stability of wave function overlaps is carried out, and strategies for overcoming potential severe pitfalls due to displaced atoms and truncated wave functions are presented.
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      Malmqvist, P.-Å.; Roos, B. O. The CASSCF State Interaction Method. Chem. Phys. Lett. 1989, 155, 189– 194,  DOI: 10.1016/0009-2614(89)85347-3
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      The CAS-SCF state interaction method
      Malmqvist, Per Aake; Roos, Bjoern O.
      Chemical Physics Letters (1989), 155 (2), 189-94CODEN: CHPLBC; ISSN:0009-2614.
      A method is described to calc. the matrix elements of one- and two-electron operators for CASSCF wave functions employing individually optimized orbitals. The computation procedure is very efficient, and matrix elements between CAS wave functions with up to a few hundred thousand configurations are easily evaluated. An implementation of this method is presented, which sets up and solves the Hamiltonian secular problem in a basis of independently optimized CASSCF wave functions. In the program, the only restriction imposed is that the AO basis set and the no. of active orbitals is the same for all states. The method is illustrated by calcns. on π-excited states of some arom. mols.
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      Malmqvist, P. k. Calculation of Transition Density Matrices by Nonunitary Orbital Transformations. Int. J. Quantum Chem. 1986, 30, 479– 494,  DOI: 10.1002/qua.560300404
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      Calculation of transition density matrixes by nonunitary orbital transformations
      Malmqvist, Per Aake
      International Journal of Quantum Chemistry (1986), 30 (4), 479-94CODEN: IJQCB2; ISSN:0020-7608.
      An efficient scheme for calcg. one- and two-electron transition d. matrixes for 2 wave functions is described. The method applies to complete active space wave functions and certain multireference CI expansions. The orbital sets of the 2 wave functions are not assumed to be equal. They are transformed to a biorthonormal basis, and the corresponding transformation of the CI coeffs. is carried out directly, using the one-electron coupling coeffs.
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      Olsen, J.; Godefroid, M. R.; Jönsson, P.; Malmqvist, P. Å.; Fischer, C. F. Transition Probability Calculations for Atoms Using Nonorthogonal Orbitals. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1995, 52, 4499– 4508,  DOI: 10.1103/physreve.52.4499
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      Transition probability calculations for atoms using nonorthogonal orbitals
      Olsen, Jeppe; Godefroid, Michel R.; Joensson, Per; Malmqvist, Per Ake; Fischer, Charlotte Froese
      Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics (1995), 52 (4-B), 4499-508CODEN: PLEEE8; ISSN:1063-651X. (American Physical Society)
      Individual orbital optimization of wave functions for the initial and final states produces the most accurate wave functions for given expansions, but complicates the calcn. of transition-matrix elements since the two sets of orbitals will be nonorthogonal. The orbital sets can be transformed to become biorthonormal, in which case the evaluation of any matrix element can proceed as in the orthonormal case. The transformation of the wave-function expansion to the new basis imposes certain requirements on the wave function, depending on the type of transformation. An efficient and general method was found a few years ago for expansions in determinants, spin-coupled configurations, or configuration state functions for mols. belonging to the D2h point group or its subgroups. The method requires only that the expansions are closed under deexcitation and thus applies to restricted active space wave functions. This type of expansion is efficient for correlation studies and includes many types of expansions as special cases. The above technique has been generalized to the at., symmetry adapted case requiring the treatment of degenerate shells nlN, with arbitrary occupation nos. 0 ≤ N ≤ 4l + 2. A computer implementation of the algorithm in the multiconfiguration Hartree-Fock at.-structure package for atoms allows the calcn. of transition moments for individually optimized states. An application is presented for the BI 1s22s12p2Po → 1s22s2p22D elec. dipole transition probability, which is highly sensitive to core-polarization effects.
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      Malmqvist, P. A.; Rendell, A.; Roos, B. O. The Restricted Active Space Self-Consistent-Field Method, Implemented with a Split Graph Unitary Group Approach. J. Phys. Chem. 1990, 94, 5477– 5482,  DOI: 10.1021/j100377a011
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      The restricted active space self-consistent-field method, implemented with a split graph unitary group approach
      Malmqvist, Per Aake; Rendell, Alistair; Roos, Bjoern O.
      Journal of Physical Chemistry (1990), 94 (14), 5477-82CODEN: JPCHAX; ISSN:0022-3654.
      An MCSCF method based on a restricted active space (RAS) type wave function has been implemented. The RAS concept is an extension of the complete active space (CAS) formalism, where the active orbitals are partitioned into three subspaces: RAS1, which contains up to a given max. no. of holes; RAS2, where all possible distributions of electrons are allowed; and RAS3, which contains up to a given max. no. of electrons. A typical example of a RAS wave function is all single, double, etc. excitations with a CAS ref. space. Spin-adapted configurations are used as the basis for the MC expansion. Vector coupling coeffs. are generated by using a split graph variant of the unitary group approach, with the result that very large MC expansions (up to around 106) can be treated, still keeping the no. of coeffs. explicitly computed small enough to be kept in central storage. The largest MCSCF calcn. that has so far been performed with the new program is for a CAS wave function contg. 716,418 CSF's. The MCSCF optimization is performed using an approx. form of the super-CI method. A quasi-Newton update method is used as a convergence accelerator and results in much improved convergence in most applications.
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      Knecht, S.; Keller, S.; Autschbach, J.; Reiher, M. A Nonorthogonal State-Interaction Approach for Matrix Product State Wave Functions. J. Chem. Theory Comput. 2016, 12, 5881– 5894,  DOI: 10.1021/acs.jctc.6b00889
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      A Nonorthogonal State-Interaction Approach for Matrix Product State Wave Functions
      Knecht, Stefan; Keller, Sebastian; Autschbach, Jochen; Reiher, Markus
      Journal of Chemical Theory and Computation (2016), 12 (12), 5881-5894CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present a state-interaction approach for matrix product state (MPS) wave functions in a nonorthogonal MO basis. Our approach allows us to calc. for example transition and spin-orbit coupling matrix elements between arbitrary electronic states provided that they share the same one-electron basis functions and active orbital space, resp. The key element is the transformation of the MPS wave functions of different states from a nonorthogonal to a biorthonormal MO basis representation exploiting a sequence of non-unitary transformations following a proposal by Malmqvist (Int. J. Quantum Chem.30, 479 (1986)). This is well-known for traditional wave-function parametrizations but has not yet been exploited for MPS wave functions.
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      Roemelt, M. Spin Orbit Coupling for Molecular Ab Initio Density Matrix Renormalization Group Calculations: Application to g-Tensors. J. Chem. Phys. 2015, 143, 044112,  DOI: 10.1063/1.4927432
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      Spin orbit coupling for molecular ab initio density matrix renormalization group calculations: Application to g-tensors
      Roemelt, Michael
      Journal of Chemical Physics (2015), 143 (4), 044112/1-044112/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Spin Orbit Coupling (SOC) is introduced to mol. ab initio d. matrix renormalization group (DMRG) calcns. In the presented scheme, one 1st approximates the electronic ground state and a no. of excited states of the Born-Oppenheimer (BO) Hamiltonian with the aid of the DMRG algorithm. Owing to the spin-adaptation of the algorithm, the total spin S is a good quantum no. for these states. After the nonrelativistic DMRG calcn. is finished, all magnetic sublevels of the calcd. states are constructed explicitly, and the SOC operator is expanded in the resulting basis. To this end, spin orbit coupled energies and wavefunctions were obtained as eigenvalues and eigenfunctions of the full Hamiltonian matrix which is composed of the SOC operator matrix and the BO Hamiltonian matrix. This treatment corresponds to a quasi-degenerate perturbation theory approach and can be regarded as the mol. equiv. to at. Russell-Saunders coupling. For the evaluation of SOC matrix elements, the full Breit-Pauli SOC Hamiltonian is approximated by the widely used spin-orbit mean field operator. This operator allows for an efficient use of the 2nd quantized triplet replacement operators that are readily generated during the nonrelativistic DMRG algorithm, together with the Wigner-Eckart theorem. With a set of spin-orbit coupled wavefunctions at hand, the mol. g-tensors are calcd. following the scheme proposed by Gerloch and McMeeking. It interprets the effective mol. g-values as the slope of the energy difference between the lowest Kramers pair with respect to the strength of the applied magnetic field. Test calcns. on a chem. relevant Mo complex demonstrate the capabilities of the presented method. (c) 2015 American Institute of Physics.
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      Sayfutyarova, E. R.; Chan, G. K.-L. A State Interaction Spin-Orbit Coupling Density Matrix Renormalization Group Method. J. Chem. Phys. 2016, 144, 234301,  DOI: 10.1063/1.4953445
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      A state interaction spin-orbit coupling density matrix renormalization group method
      Sayfutyarova, Elvira R.; Chan, Garnet Kin-Lic
      Journal of Chemical Physics (2016), 144 (23), 234301/1-234301/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The authors describe a state interaction spin-orbit (SISO) coupling method using d. matrix renormalization group (DMRG) wavefunctions and the spin-orbit mean-field (SOMF) operator. The authors implement the authors' DMRG-SISO scheme using a spin-adapted algorithm that computes transition d. matrixes between arbitrary matrix product states. To demonstrate the potential of the DMRG-SISO scheme the authors present accurate benchmark calcns. for the zero-field splitting of the copper and gold atoms, comparing to earlier complete active space SCF and 2nd-order complete active space perturbation theory results in the same basis. The authors also compute the effects of spin-orbit coupling on the spin-ladder of the iron-sulfur dimer complex [Fe2S2(SCH3)4]3-, detg. the splitting of the lowest quartet and sextet states. The magnitude of the zero-field splitting for the higher quartet and sextet states approaches a significant fraction of the Heisenberg exchange parameter. (c) 2016 American Institute of Physics.
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      Sayfutyarova, E. R.; Chan, G. K.-L. Electron Paramagnetic Resonance G-Tensors from State Interaction Spin-Orbit Coupling Density Matrix Renormalization Group. J. Chem. Phys. 2018, 148, 184103,  DOI: 10.1063/1.5020079
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      Electron paramagnetic resonance g-tensors from state interaction spin-orbit coupling density matrix renormalization group
      Sayfutyarova, Elvira R.; Chan, Garnet Kin-Lic
      Journal of Chemical Physics (2018), 148 (18), 184103/1-184103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present a state interaction spin-orbit coupling method to calc. ESR g-tensors from d. matrix renormalization group wavefunctions. We apply the technique to compute g-tensors for the TiF3 and CuCl2-4 complexes, a [2Fe-2S] model of the active center of ferredoxins, and a Mn4CaO5 model of the S2 state of the oxygen evolving complex. These calcns. raise the prospects of detg. g-tensors in multireference calcns. with a large no. of open shells. (c) 2018 American Institute of Physics.
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      Knecht, S.; Legeza, Ö.; Reiher, M. Communication: Four-Component Density Matrix Renormalization Group. J. Chem. Phys. 2014, 140, 041101,  DOI: 10.1063/1.4862495
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      Communication. Four-component density matrix renormalization group
      Knecht, Stefan; Legeza, Ors; Reiher, Markus
      Journal of Chemical Physics (2014), 140 (4), 041101/1-041101/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present the 1st implementation of the relativistic quantum chem. 2- and 4-component d. matrix renormalization group algorithm that includes a variational description of scalar-relativistic effects and spin-orbit coupling. Numerical results based on the 4-component Dirac-Coulomb Hamiltonian are presented for the std. ref. mol. for correlated relativistic benchmarks: TlH. (c) 2014 American Institute of Physics.
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      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.7b01065
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      Efficient Relativistic Density-Matrix Renormalization Group Implementation in a Matrix-Product Formulation
      Battaglia, Stefano; Keller, Sebastian; Knecht, Stefan
      Journal 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.
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    30. 30
      Crespo-Otero, R.; Barbatti, M. Recent Advances and Perspectives on Nonadiabatic Mixed Quantum-Classical Dynamics. Chem. Rev. 2018, 118, 7026– 7068,  DOI: 10.1021/acs.chemrev.7b00577
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      Recent Advances and Perspectives on Nonadiabatic Mixed Quantum-Classical Dynamics
      Crespo-Otero, Rachel; Barbatti, Mario
      Chemical Reviews (Washington, DC, United States) (2018), 118 (15), 7026-7068CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
      A review. Nonadiabatic mixed quantum-classical (NA-MQC) dynamics methods form a class of computational theor. approaches in quantum chem. tailored to investigate the time evolution of nonadiabatic phenomena in mols. and supramol. assemblies. NA-MQC is characterized by a partition of the mol. system into two subsystems: one to be treated quantum mech. (usually but not restricted to electrons) and another to be dealt with classically (nuclei). The two subsystems are connected through nonadiabatic couplings terms to enforce self-consistency. A local approxn. underlies the classical subsystem, implying that direct dynamics can be simulated, without needing precomputed potential energy surfaces. The NA-MQC split allows reducing computational costs, enabling the treatment of realistic mol. systems in diverse fields. Starting from the three most well-established methods - mean-field Ehrenfest, trajectory surface hopping, and multiple spawning - this review focuses on the NA-MQC dynamics methods and programs developed in the last 10 years. It stresses the relations between approaches and their domains of application. The electronic structure methods most commonly used together with NA-MQC dynamics are reviewed as well. The accuracy and precision of NA-MQC simulations are critically discussed, and general guidelines to choose an adequate method for each application are delivered.
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      Lengsfield, B. H., III; Saxe, P.; Yarkony, D. R. On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wave functions and analytic gradient methods. I. J. Chem. Phys. 1984, 81, 4549– 4553,  DOI: 10.1063/1.447428
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      On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wave functions and analytic gradient methods. I
      Lengsfield, Byron H., III; Saxe, Paul; Yarkony, David R.
      Journal of Chemical Physics (1984), 81 (10), 4549-53CODEN: JCPSA6; ISSN:0021-9606.
      A method for the efficient evaluation of nonadiabatic coupling matrix elements of the form 〈ΨI|.vdelta./.vdelta.RαΨJ〉 is presented. The wave function ΨI and ΨJ are assumed to be MCSCF wave functions optimized within the state averaged (SA) approxn. The method, which can treat several states simultaneously, derives its efficiency from the direct soln. of the coupled perturbed state averaged MCSCF equations and the availability of other appropriate deriv. integrals. An extension of this approach to SA-MCSCF/CI wave functions is described. Computational efficiencies can be achieved by exploiting analogies with analytic CI gradient methods. Numerical examples for C2v approach of Mg to H2 are presented.
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      Stålring, J.; Bernhardsson, A.; Lindh, R. Analytical Gradients of a State Average MCSCF State and a State Average Diagnostic. Mol. Phys. 2001, 99, 103– 114,  DOI: 10.1080/002689700110005642
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      Analytical gradients of a state average MCSCF state and a state average diagnostic
      Stalring, Jonna; Bernhardsson, Anders; Lindh, Roland
      Molecular Physics (2001), 99 (2), 103-114CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      An efficient method for calcg. the Lagrange multipliers and the anal. gradients of one state included in a state av. MCSCF wave function is presented. The state av. energy of an "equal-wt." scheme is invariant to rotations within the state av. subspace and that the corresponding rotations should be eliminated from the Lagrangian equations. A diagnostic is presented, which gauges the energy difference between a state defined by a state av. calcn. and the corresponding fully variational multi-configurational SCF state.
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      Snyder, J. W.; Hohenstein, E. G.; Luehr, N.; Martínez, T. J. An Atomic Orbital-Based Formulation of Analytical Gradients and Nonadiabatic Coupling Vector Elements for the State-Averaged Complete Active Space Self-Consistent Field Method on Graphical Processing Units. J. Chem. Phys. 2015, 143, 154107,  DOI: 10.1063/1.4932613
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      An atomic orbital-based formulation of analytical gradients and nonadiabatic coupling vector elements for the state-averaged complete active space self-consistent field method on graphical processing units
      Snyder, James W.; Hohenstein, Edward G.; Luehr, Nathan; Martinez, Todd J.
      Journal of Chemical Physics (2015), 143 (15), 154107/1-154107/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We recently presented an algorithm for state-averaged complete active space SCF (SA-CASSCF) orbital optimization that capitalizes on sparsity in the AO basis set to reduce the scaling of computational effort with respect to mol. size. Here, we extend those algorithms to calc. the analytic gradient and nonadiabatic coupling vectors for SA-CASSCF. Combining the low computational scaling with acceleration from graphical processing units allows us to perform SA-CASSCF geometry optimizations for mols. with more than 1000 atoms. The new approach will make minimal energy conical intersection searches and nonadiabatic dynamics routine for mol. systems with O(102) atoms. (c) 2015 American Institute of Physics.
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      Snyder, J. W.; Fales, B. S.; Hohenstein, E. G.; Levine, B. G.; Martínez, T. J. A Direct-Compatible Formulation of the Coupled Perturbed Complete Active Space Self-Consistent Field Equations on Graphical Processing Units. J. Chem. Phys. 2017, 146, 174113,  DOI: 10.1063/1.4979844
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      A direct-compatible formulation of the coupled perturbed complete active space self-consistent field equations on graphical processing units
      Snyder, James W.; Fales, B. Scott; Hohenstein, Edward G.; Levine, Benjamin G.; Martinez, Todd J.
      Journal of Chemical Physics (2017), 146 (17), 174113/1-174113/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We recently developed an algorithm to compute response properties for the state-averaged complete active space SCF method (SA-CASSCF) that capitalized on sparsity in the AO basis. Our original algorithm was limited to treating small to moderate sized active spaces, but the recent development of graphical processing unit (GPU) based direct-CI algorithms provides an opportunity to extend this to large active spaces. We present here a direct-compatible version of the coupled perturbed equations, enabling us to compute response properties for systems treated with arbitrary active spaces (subject to available memory and computation time). This work demonstrates that the computationally demanding portions of the SA-CASSCF method can be formulated in terms of seven fundamental operations, including Coulomb and exchange matrix builds and their derivs., as well as, generalized one- and two-particle d. matrix and σ vector constructions. As in our previous work, this algorithm exhibits low computational scaling and is accelerated by the use of GPUs, making possible optimizations and nonadiabatic dynamics on systems with O(1000) basis functions and O(100) atoms, resp. (c) 2017 American Institute of Physics.
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      Mai, S.; Marquetand, P.; González, L. Nonadiabatic Dynamics: The SHARC Approach. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018, 8, e1370  DOI: 10.1002/wcms.1370
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      Granucci, G.; Persico, M.; Toniolo, A. Direct Semiclassical Simulation of Photochemical Processes with Semiempirical Wave Functions. J. Chem. Phys. 2001, 114, 10608– 10615,  DOI: 10.1063/1.1376633
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      Direct semiclassical simulation of photochemical processes with semiempirical wave functions
      Granucci, G.; Persico, M.; Toniolo, A.
      Journal of Chemical Physics (2001), 114 (24), 10608-10615CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The authors describe a new method for the simulation of excited state dynamics, based on classical trajectories and surface hopping, with direct semiempirical calcn. of the electronic wave functions and potential energy surfaces (DTSH method). Semiempirical self-consistent-field MOs (SCF MO's) are computed with geometry-dependent occupation nos., in order to ensure correct homolytic dissocn., fragment orbital degeneracy, and partial optimization of the lowest virtuals. Electronic wave functions are of the MO active space CI type, for which analytic energy gradients have been implemented. The time-dependent electronic wave function is propagated by means of a local diabatization algorithm which is inherently stable also in the case of surface crossings. The method is tested for the problem of excited ethylene nonadiabatic dynamics, and the results are compared with recent quantum mech. calcns.
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      Freitag, L.; Ma, Y.; Baiardi, A.; Knecht, S.; Reiher, M. Approximate Analytical Gradients and Nonadiabatic Couplings for the State-Average Density Matrix Renormalization Group Self-Consistent-Field Method. J. Chem. Theory Comput. 2019, 15, 6724– 6737,  DOI: 10.1021/acs.jctc.9b00969
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      Approximate Analytical Gradients and Nonadiabatic Couplings for the State-Average Density Matrix Renormalization Group Self-Consistent-Field Method
      Freitag, Leon; Ma, Yingjin; Baiardi, Alberto; Knecht, Stefan; Reiher, Markus
      Journal of Chemical Theory and Computation (2019), 15 (12), 6724-6737CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present an approx. scheme for anal. gradients and nonadiabatic couplings for a state-av. d. matrix renormalization group self-consistent-field (SA-DMRG-SCF) wavefunction. Our formalism follows closely the state-av. complete active space self-consistent-field (SA-CASSCF) ansatz, which employs a Lagrangian, and the corresponding Lagrange multipliers are obtained from a soln. of the coupled-perturbed CASSCF (CP-CASSCF) equations. We introduce a definition of the MPS Lagrange multipliers based on a single-site tensor in a mixed-canonical form of the MPS, such that the sweep procedure is avoided in the soln. of the CP-CASSCF equations. We employ our implementation for the optimization of a conical intersection in 1,2-dioxetanone, where we are able to fully reproduce the SA-CASSCF result up to arbitrary accuracy.
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      Schollwöck, U. The Density-Matrix Renormalization Group in the Age of Matrix Product States. Ann. Phys. 2011, 326, 96– 192,  DOI: 10.1016/j.aop.2010.09.012
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      Schollwöck, U. The Density-Matrix Renormalization Group: A Short Introduction. Philos. Trans. R. Soc., A 2011, 369, 2643– 2661,  DOI: 10.1098/rsta.2010.0382
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      Keller, S.; Dolfi, M.; Troyer, M.; Reiher, M. An Efficient Matrix Product Operator Representation of the Quantum Chemical Hamiltonian. J. Chem. Phys. 2015, 143, 244118,  DOI: 10.1063/1.4939000
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      An efficient matrix product operator representation of the quantum chemical Hamiltonian
      Keller, Sebastian; Dolfi, Michele; Troyer, Matthias; Reiher, Markus
      Journal of Chemical Physics (2015), 143 (24), 244118/1-244118/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We describe how to efficiently construct the quantum chem. Hamiltonian operator in matrix product form. We present its implementation as a d. matrix renormalization group (DMRG) algorithm for quantum chem. applications. Existing implementations of DMRG for quantum chem. are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the addnl. flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - Abelian and non-Abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program. (c) 2015 American Institute of Physics.
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      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: Condens. Matter Mater. Phys. 2003, 67, 125114,  DOI: 10.1103/physrevb.67.125114
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      Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach
      Legeza, 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.
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      Ronconi, L.; Sadler, P. J. Photoreaction pathways for the anticancer complex trans,trans,trans-[Pt(N3)2(OH)2(NH3)2]. Dalton Trans. 2011, 40, 262– 268,  DOI: 10.1039/c0dt00546k
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      Photoreaction pathways for the anticancer complex trans,trans,trans-[Pt(N3)2(OH)2(NH3)2]
      Ronconi, Luca; Sadler, Peter J.
      Dalton Transactions (2011), 40 (1), 262-268CODEN: DTARAF; ISSN:1477-9226. (Royal Society of Chemistry)
      The photodecompn. of the anticancer complex trans,trans,trans-[Pt(N3)2(OH)2(NH3)2] in acidic aq. soln., as well as in phosphate-buffered saline (PBS), induced by UVA light (centered at λ = 365 nm) has been studied by multinuclear NMR spectroscopy. The authors show that the photoreaction pathway in PBS, which involves azide release, differs from that in acidic aq. conditions, under which N2 is a major product. In both cases, a no. of trans-{N-Pt(II/IV)-NH3} species were also obsd. as photoproducts, as well as the evolution of O2 and release of free ammonia with a subsequent increase in pH. The results from this study illustrate that photoinduced reactions of Pt(IV)-diazido derivs. can lead to novel reaction pathways, and therefore potentially to new cytotoxic mechanisms in cancer cells.
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      Bednarski, P. J.; Korpis, K.; Westendorf, A. F.; Perfahl, S.; Grünert, R. Effects of light-activated diazido-Pt IV complexes on cancer cells in vitro. Philos. Trans. R. Soc., A 2013, 371, 20120118,  DOI: 10.1098/rsta.2012.0118
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      Effects of light-activated diazido-PtIV complexes on cancer cells in vitro
      Bednarski, Patrick J.; Korpis, Katharina; Westendorf, Aron F.; Perfahl, Steffi; Gruenert, Renate
      Philosophical Transactions of the Royal Society, A: Mathematical, Physical & Engineering Sciences (2013), 371 (1995), 20120118/1-20120118/15CODEN: PTRMAD; ISSN:1364-503X. (Royal Society)
      A review. Various PtIV diazides have been investigated over the years as light-activatable prodrugs that interfere with cell proliferation, accumulate in cancer cells and cause cell death. The potencies of the complexes vary depending on the substituted amines (pyridine = piperidine > ammine) as well as the coordination geometry (trans diazide > cis). Light-activated PtIV diazides tend to be less specific than cisplatin at inhibiting cancer cell growth, but cells resistant to cisplatin show little cross-resistance to PtIV diazides. Platinum is accumulated in the cancer cells to a similar level as cisplatin, but only when activated by light, indicating that reactive Pt species form photolytically. Studies show that Pt also becomes attached to cellular DNA upon the light activation of various PtIV diazides. Structures of some of the photolysis products were elucidated by LC-MS/MS; monoaqua- and diaqua-PtII complexes form that are reactive towards biomols. such as calf thymus DNA. Platination of calf thymus DNA can be blocked by the addn. of nucleophiles such as glutathione and chloride, further evidence that aqua-PtII species form upon irradn. Evidence is presented that reactive oxygen species may be generated in the first hours following photoactivation. Cell death does not take the usual apoptotic pathways seen with cisplatin, but appears to involve autophagy. Thus, photoactivated diazido-PtIV complexes represent an interesting class of potential anti-cancer agents that can be selectively activated by light and kill cells by a mechanism different to the anti-cancer drug cisplatin.
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      Shi, H.; Imberti, C.; Sadler, P. J. Diazido Platinum(IV) Complexes for Photoactivated Anticancer Chemotherapy. Inorg. Chem. Front. 2019, 6, 1623– 1638,  DOI: 10.1039/c9qi00288j
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      Diazido platinum(IV) complexes for photoactivated anticancer chemotherapy
      Shi, Huayun; Imberti, Cinzia; Sadler, Peter J.
      Inorganic Chemistry Frontiers (2019), 6 (7), 1623-1638CODEN: ICFNAW; ISSN:2052-1553. (Royal Society of Chemistry)
      A review. Diazido Pt(IV) complexes with a general formula [Pt(N3)2(L)(L')(OR)(OR')] are a new generation of anticancer prodrugs designed for use in photoactivated chemotherapy. The potencies of these complexes are affected by the cis/trans geometry configuration, the non-leaving ligand L/L' and derivatisation of the axial ligand OR/OR'. Diazido Pt(IV) complexes exhibit high dark stability and promising photocytotoxicity circumventing cisplatin resistance. Upon irradn., diazido Pt(IV) complexes release anticancer active Pt(II) species, azidyl radicals and ROS, which interact with biomols. and therefore affect the cellular components and pathways. The conjugation of diazido Pt(IV) complexes with anticancer drugs or cancer-targeting vectors is an effective strategy to optimize the design of these photoactive prodrugs. Diazido Pt(IV) complexes represent a series of promising anticancer prodrugs owing to their novel mechanism of action which differs from that of classical cisplatin and its analogs.
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    45. 45
      Freitag, L.; González, L. The Role of Triplet States in the Photodissociation of a Platinum Azide Complex by a Density Matrix Renormalization Group Method. J. Phys. Chem. Lett. 2021, 12, 4876– 4881,  DOI: 10.1021/acs.jpclett.1c00829
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      45
      The Role of Triplet States in the Photodissociation of a Platinum Azide Complex by a Density Matrix Renormalization Group Method
      Freitag, Leon; Gonzalez, Leticia
      Journal of Physical Chemistry Letters (2021), 12 (20), 4876-4881CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      Platinum azide complexes are appealing anticancer photochemotherapy drug candidates because they release cytotoxic azide radicals upon light irradn. Here we present a d. matrix renormalization group SCF (DMRG-SCF) study of the azide photodissocn. mechanism of trans,trans,trans-[Pt(N3)2(OH)2(NH3)2], including spin-orbit coupling. We find a complex interplay of singlet and triplet electronic excited states that falls into three different dissocn. channels at well-sepd. energies. These channels can be accessed either via direct excitation into barrierless dissociative states or via intermediate doorway states from which the system undergoes non-radiative internal conversion and intersystem crossing. The high d. of states, particularly of spin-mixed states, is key to aid non-radiative population transfer and enhance photodissocn. along the lowest electronic excited states.
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      Hammes-Schiffer, S.; Tully, J. C. Proton Transfer in Solution: Molecular Dynamics with Quantum Transitions. J. Chem. Phys. 1994, 101, 4657,  DOI: 10.1063/1.467455
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      Proton transfer in solution: molecular dynamics with quantum transitions
      Hammes-Schiffer, Sharon; Tully, John C.
      Journal of Chemical Physics (1994), 101 (6), 4657-67CODEN: JCPSA6; ISSN:0021-9606.
      We apply mol. dynamics with quantum transitions (MDQT), a surface-hopping method previously used only for electronic transitions, to proton transfer in soln., where the quantum particle is an atom. We use full classical mech. mol. dynamics for the heavy atom degrees of freedom, including the solvent mols., and treat the hydrogen motion quantum mech. We identify new obstacles that arise in this application of MDQT and present methods for overcoming them. We implement these new methods to demonstrate that application of MDQT to proton transfer in soln. is computationally feasible and appears capable of accurately incorporating quantum mech. phenomena such as tunneling and isotope effects. As an initial application of the method, we employ a model used previously by Azzouz and Borgis (1993) to represent the proton transfer reaction AH-B .dblharw. A--H+B in liq. Me chloride, where the AH-B complex corresponds to a typical phenol-amine complex. We have chosen this model, in part, because it exhibits both adiabatic and diabatic behavior, thereby offering a stringent test of the theory. MDQT proves capable of treating both limits, as well as the intermediate regime. Up to four quantum states were included in this simulation, and the method can easily be extended to include addnl. excited states, so it can be applied to a wide range of processes, such as photoassisted tunneling. In addn., this method is not perturbative, so trajectories can be continued after the barrier is crossed to follow the subsequent dynamics.
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      Plasser, F.; Granucci, G.; Pittner, J.; Barbatti, M.; Persico, M.; Lischka, H. Surface Hopping Dynamics Using a Locally Diabatic Formalism: Charge Transfer in the Ethylene Dimer Cation and Excited State Dynamics in the 2-Pyridone Dimer. J. Chem. Phys. 2012, 137, 22A514,  DOI: 10.1063/1.4738960
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      Plasser, F.; González, L. Communication: Unambiguous Comparison of Many-Electron Wavefunctions through Their Overlaps. J. Chem. Phys. 2016, 145, 021103,  DOI: 10.1063/1.4958462
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      Communication: Unambiguous comparison of many-electron wavefunctions through their overlaps
      Plasser, Felix; Gonzalez, Leticia
      Journal of Chemical Physics (2016), 145 (2), 021103/1-021103/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A simple and powerful method for comparing many-electron wavefunctions constructed at different levels of theory is presented. By using wavefunction overlaps, it is possible to analyze the effects of varying wavefunction models, MOs, and one-electron basis sets. The computation of wavefunction overlaps eliminates the inherent ambiguity connected to more rudimentary wavefunction anal. protocols, such as visualization of orbitals or comparing selected phys. observables. Instead, wavefunction overlaps allow processing the many-electron wavefunctions in their full inherent complexity. The presented method is particularly effective for excited state calcns. as it allows for automatic monitoring of changes in the ordering of the excited states. A numerical demonstration based on multireference computations of two test systems, the selenoacrolein mol. and an iridium complex, is presented. (c) 2016 American Institute of Physics.
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  • Supporting Information

    Supporting Information

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    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.1c00674.

    • Computational details and details regarding the active space orbitals (PDF)


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