On the potentially transformative role of auxiliary-field quantum Monte Carlo in quantum chemistry: A highly accurate method for transition metals and beyond

James Shee, John L. Weber, David R. Reichman, Richard A. Friesner, and Shiwei Zhang (2022)
Highlighted by Jan Jensen

Figure 1 from this paper. (c) the authors

This paper highlights a big problem in the field of quantum chemistry and posits that a solution may be right around the corner. The problem is that we still can't routinely predict the thermochemistry of TM-containing compounds with the same degree of accuracy as we can for organic molecules. The main reason is that the former systems often have a high-degree of non-dynamic correlation which means that our CCSD(T) often does not give reliable results. We can model the non-dynamic correlation with CASSCF, but there is no good way to compute the dynamic correlation based on a CASSCF wavefunction. So when different DFT functional results give wildly different predictions for your TM-compound there is no way to tell which method, if any, if the best.

This paper argues that phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) may be the solution to this problem. ph-AFQMC represents the ground state as a stochastic linear combination of Slater determinants mapped as open-ended random walks starting from a trial wavefunction. The method accounts for both non-dynamic and dynamic correlation and the paper argues that chemical accuracy can be achieved with a few hundred random walks, which can be run in parallel and on GPUs.

So what's missing? According to the authors some of the improvements needed include: more efficient ways of reaching the CBS limit, more efficient random walks and a general, automatable protocol to generate optimal trial wave functions. Let's hope these improvements will be made soon, so we can explore a much larger portion of chemical space with confidence.

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The minimum parameterization of the wave function for the many-body electronic Schrödinger equation. I. Theory and ansatz

Lasse Kragh Sørensen (2019)
Highlighted by Jan Jensen

It is well known that Full CI (FCI) scales exponentially with the basis set size. However, Sørensen claims that the "whole notion of the exponentially scaling nature of quantum mechanics simply stems from expanding the wavefunction in a sub-optimal basis",  i.e. one-electron functions. Sørensen goes on to argue that of two-ele tron functions (geminals) are used instead, the scaling is reduced to m(m−1)/2 where m is the number of basis functions.  Furthermore, because "the number of parameters is independent of the electronic structure the multi-configurational problem is a mere phantom summoned by a poor choice of basis for the wave function". 

I don't have the background to tell whether the arguments in this paper are correct and the main point os this post is to see if I can get some feedback from people who do have the background. 

In principle one could simply compare the new approach to FCI calculations but the new method isn't quite there yet:

A straight forward minimization of Eq. 84 unfortunately gives the solution for the two-electron problem of a +(N−2) charged system N/2 times so additional constraints must be introduced. These constraints can be found by the property of the wave function in limiting cases. The problem of finding the constraints for the geminals in the AGP ansatz is therefore closely related to finding the N-representability constraints for the two-body reduced density matrix (2-RDM). For the N-representability Mazziotti have showed a constructive solution[74] though the exact conditions are still illusive.[75, 76]  

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A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects

Grimme, S.; Hansen, A. Angew. Chem. Int. Ed. 2015, 54, 12308-12313
Contributed by Steven Bachrach
Reposted from Computational Organic Chemistry with permission

Assessing when a molecular system might be subject to sizable static (non-dynamic) electron correlation, necessitating a multi-reference quantum mechanical treatment, is perhaps more art than science. In general one suspects that static correlation will be important when the frontier MO energy gap is small, but is there a way to get more guidance?

Grimme reports the use of fractional occupancy density (FOD) as a visualization tool to identify regions within molecules that demonstrate significant static electron correlation.¹ The method is based on the use of finite temperature DFT.^2,3

The resulting plots of the FOD for a series of test cases follow our notions of static correlation. Molecules, such as alkanes, simple aromatics, and concerted transition states show essentially no fractional orbital density. On the other hand, the FOD plot for ozone shows significant density spread over the entire molecule; the transition state for the cleavage of the terminal C-C bond in octane shows FOD at C₁ and C₂but not elsewhere; p-benzyne shows significant FOD at the two radical carbons, while the FOD is much smaller in m-benzyne and is negligible in o-benzyne.

This FOD method looks to be a simple tool for evaluating static correlation and is worth further testing.

References

(1) Grimme, S.; Hansen, A. "A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects," Angew. Chem. Int. Ed. 2015, 54, 12308-12313, DOI: 10.1002/anie.201501887.

(2) Mermin, N. D. "Thermal Properties of the Inhomogeneous Electron Gas," Phys. Rev. 1965, 137, A1441-A1443, DOI: 10.1103/PhysRev.137.A1441.

(3) Chai, J.-D. "Density functional theory with fractional orbital occupations," J. Chem. Phys 2012, 136, 154104, DOI: doi: 10.1063/1.3703894.

[Editors note: this paper has also been highlighted by Tobias Schwabe]


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Stochastic Multiconfigurational Self-Consistent Field Theory

Robert E. Thomas, Qiming Sun, Ali Alavi,  & George H. Booth, (2015) DOI: 10.1021/acs.jctc.5b00917
Contributed by Jan Jensen

The multiconfigurational self-consistent field (MCSCF), especially complete active space SCF (CASSCF)  method remains the only systematically improveable option for a host of molecules and processes with strong non-dynamical correlation.  However, the requirement of a full configuration interaction (FCI) calculation for the active space results in a computational cost that scales exponentially with the size of the active space.  There are therefore many interesting problems that remain out of reach for CASSCF. But perhaps not for long.

This papers shows that the FCI steps of the CASSCF calculation can be replaced by FCI quantum Monte Carlo (FCIQMC) calculations, which scales significantly better than FCI.  In the FCIQMC approach new determinants ($j$) are generated from old determinants ($i$) stochastically and included if the probability of this generation step exceeds a randomly chosen number between 0 and 1. The authors show that ca 25,000 sampling iterations yield energies within 0.1 milli-Hartrees of conventional CASSCF. As a result this approach could be used for a complete active space with 24 electrons in 24 orbitals with "only modest computational resources"!

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Can Density Cumulant Functional Theory Describe Static Correlation Effects?

Mullinax, J. W.; Sokolov, A. Y.; Schaefer, H. F. J. Chem. Theor. Comput. 2015, 11, 2487-2495 
Contributed by Steven Bachrach.
Reposted from Computational Organic Chemistry with permission

I want to update my discussion of m-benzyne, which I present in my book in Chapter 5.5.3. The interesting question concerning m-benzyne concerns its structure: is it a single ring structure 1a or a bicyclic structure 1b? Single configuration methods including closed-shell DFT methods predict the bicylic structure, but multi-configuration methods and unrestricted DFT predict it to be 1a. Experiments support the single ring structure 1a.

The key measurement distinguishing these two structure type is the C₁-C₃ distance. Table 1 updates Table 5.11 from my book with the computed value of this distance using some new methods. In particular, the state-specific multireference coupled cluster Mk-MRCCSD method¹ with the cc-pCVTZ basis set indicates a distance of 2.014 Å.² The density cumulant functional theory³ ODC-12⁴ with the cc-pCVTZ basis set also predicts the single ring structure with a distance of 2.101 Å.⁵

Table 1. C₁-C₃ distance (Å) with different computational methods using the cc-pCVTZ basis set

method	r(C1-C3)
CCSD5	1.556
CCSD(T)5	2.043
OCD-125	2.101
Mk-MRCCSD2	2.014

References

(1) Evangelista, F. A.; Allen, W. D.; Schaefer III, H. F. "Coupling term derivation and general implementation of state-specific multireference coupled cluster theories," J. Chem. Phys 2007, 127, 024102-024117, DOI:10.1063/1.2743014.

(2) Jagau, T.-C.; Prochnow, E.; Evangelista, F. A.; Gauss, J. "Analytic gradients for Mukherjee’s multireference coupled-cluster method using two-configurational self-consistent-field orbitals," J. Chem. Phys. 2010, 132, 144110, DOI: 10.1063/1.3370847.

(3) Kutzelnigg, W. "Density-cumulant functional theory," J. Chem. Phys. 2006, 125, 171101, DOI:10.1063/1.2387955.

(4) Sokolov, A. Y.; Schaefer, H. F. "Orbital-optimized density cumulant functional theory," J. Chem. Phys.2013, 139, 204110, DOI: 10.1063/1.4833138.

(5) Mullinax, J. W.; Sokolov, A. Y.; Schaefer, H. F. "Can Density Cumulant Functional Theory Describe Static Correlation Effects?," J. Chem. Theor. Comput. 2015, 11, 2487-2495, DOI: 10.1021/acs.jctc.5b00346.

InChIs

1a: InChI=1S/C6H4/c1-2-4-6-5-3-1/h1-3,6H

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A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects

Grimme, S. and Hansen, A., Angew Chem Int Edit, (2015)
Contributed by Tobias Schwabe

The question of how to deal with multireference (MR) cases in DFT has a longstanding history. Of course, the exact functional would also include multireference effects (or non-dynamical/non-local/static electron correlaton, as these effects are also called) and no special care is needed. But when it comes to today's density functional approximations (DFAs) within the Kohn-Sham framework, everything is a little bit more complicated. For example, Baerends and co-workers have shown that is the exchange part in GGA-DFAs that actually accounts for static electron correlation.[1] These studies, among others, led to the conclusion that the (erroneous) electron self-interaction in DFAs accounts for some of the MR character in a system. A good review about how these things are interconnected can be found in Ref. [2].

Instead of searching for better and better DFAs, another approach to the problem is to apply ensemble DFT which introduces the free electron energy and also the concept of entropy into DFT.[3] The key concept here is to allow for fractional occupation numbers in Kohn-Sham orbitals and to look at the system at T > 0 K. In case of systems with MR character which cannot be described with a single Slater determinant fractional occupation will result (for e.g. when computing natural occupation numbers). The interesting thing about ensemble DFT is that it allows to find these numbers directly via a variational approach without computing an MR wavefunction first.

Grimme and Hansen now turned this approach into a tool for a qualitative analysis of molecular systems. They do so by plotting what they call the fractional orbital density (FOD). That is, only those molecular orbitals with non-integer occupation numbers contribute to the density – and only at a finite temperature. This density vanishes completely at T = 0 K. So, the analysis literally shows MR hot spots. Integrating the FOD yields also an absolute scalar which allows to quantify the MR character and to compare different molecules. Due to the authors, this value correlates well with other values which attempts to provide such information.

A great advantage of the approach is that now the MR character can be located (geometrically) within the molecule. The findings presented in the application part of the paper go along well with chemical intuition. The analysis might help to visualize and to interpret MR phenomenon. The tool can provide insight when the nature of the electronic structure is not obvious – for example, when dealing with biradicals in a singlet spin state. It might also be a good starting point to identify relevant regions/orbitals which should be included when one wants to treat a system on a higher level than DFT, for example with WFT-in-DFT based on projector techniques. Last but not least, it can help to identify chemical systems to which standard DFAs should not (or only with great care) be applied.

References:

[1] a) O. V. Grittsenko, P. R. T. Schipper, and E. J. Baerends, J. Chem. Phys. (1997), 107, 5007 b) P. R. T. Schipper, O. V. Grittsenko, and E. J. Baerends, Phys. Rev. A (1998), 57, 1729 c) P. R. T. Schipper, O. V. Grittsenko, and E. J. Baerends, J. Chem. Phys. (1999), 111, 4056

[2] A. J. Cohen, P. Mori-Sánchez, and W. Yang., Chem. Rev. (2012), 112, 289

[3] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press (1989)

Empirical correction of nondynamical correlation energy for density functionals

Wanyi Jiang, Chris C. Jeffrey, and Angela K. Wilson J. Phys. Chem. A 2012, 116, 9969 (Paywall)
Contributed by Grant Hill.

The inability of common density functionals to correctly account for dispersion interactions has been addressed in a number of ways, but the most popular method is to add an empirical dispersion correction to the DFT energy [1]. In this paper, Wilson and co-workers propose an empirical correction for nondynamic correlation that operates in a superficially similar way.

If one defines nondynamic correlation as the significant contribution of several electronic configurations to the total energy of a system, it can be seen that this type of correlation becomes important for a number of chemically relevant situations, including the breaking of covalent bonds. Some of the methods typically used to recover nondynamic correlation include CASSCF and MRCI, with the common theme that they quickly become expensive in terms of computational cost, and that a degree of expertise is required in the choice of which orbitals and electrons to include in the active space of nondynamic correlation. The method proposed attempts to bypass these difficulties by carrying out a standard DFT calculation, then adding a correction for nondynamic correlation (with an empirical scale factor) via a CASCI calculation including a small set of orbitals in the active space. The authors suggest that this choice of orbitals can be automated, producing a computationally efficient black-box method.

The initial results indicate that the method performs well for the torsion of ethylene and automerization of cyclobutadiene, yet when investigating barrier heights it seems that the best results are produced when only the transition state is empirically corrected. The results presented suggest that the method is worthy of further investigation, and I for one would be very interested to see if how it performs for spin-state splittings of transition metal complexes [2].

References
[1] See S. Grimme, J. Antony, S. Ehrlich, and H. Krieg J. Chem. Phys. 2010, 132, 154104 and references therein.
[2] For a perspective on spin-state splittings in bio-inorganic systems see M. Swart Int. J. Quantum Chem. 2012, DOI: 10.1002/qua.24255