# Is HF the only method for trying to solve the many-electron Schrödinger equation?

The Schrödinger equation can only be solved analytically for the smallest of "molecular" systems. The Hartree-Fock method is a method of obtaining approximate solutions to the many-electron Schrödinger equation, and a bunch of methods try to reduce the error associate with the HF method.

However, all ab initio electron correlated methods I know of and have read about are based on HF; that is, they all use the HF solution and adds to it. So basically there is only one way in use today of attempting to solve the many-electron Schrödinger equation?

Then of course there is density functional theory, but here you strictly don't try to solve the Schrödinger equation, you just try to obtain the solution, right?

So is this a correct assessment? Do we truly only use one method of trying to solve the many-electron Schrödinger equation? Is there any research being done on alternative methods (other than DFT)?

• Ab-initio valence-bond method, maybe? May 4 '16 at 16:13
• I'm not sure I'd characterize Hartree-Fock as 'solving' it either, although it is perhaps semantics to distinguish 'solving' from finding a 'solution'. If you have a solution, doesn't that solve the equation? May 4 '16 at 18:27
• There are Quantum Monte Carlo methods. May 4 '16 at 20:34
• Also, DFT does actually solve the Schroedinger equation (at least in the same sense as HF does). Especially when you look at Levi's constrained search formalism and Kohn-Sham Theory you see that you start out from the many-body Schroedinger equation and through a series of constrained minimization steps arrive at the Kohn-Sham equations. May 4 '16 at 20:43
• MCSCF is not based on HF
– user23061
May 24 '16 at 15:03

So is this a correct assessment? Do we truly only use one method of trying to solve the many-electron Schrödinger equation? Is there any research being done on alternative methods (other than DFT)?

No, it is not correct. Saying that Hartree-Fock(HF) theory is the only method is falling short. Hartree-Fock is only one of many existing methods. While it is true that some single reference methods have a dependence on the HF solution (because they build upon it) when they include a reduced number of determinants, it is not the case for other. Particular mention deserves Coupled Cluster since it is insensible to the reference determinant found by HF. The proof is given by Thouless theorem (see Joshua Goings' post, or via The Internet Archive). Thus, to give you an overview of the grand landscape of possible approaches I tried to compile the families of methods better established nowadays. I did not intend to specify individual methods because there are so many ... and people continue to investigate new ones. These are:

Deterministic

• Semi-empirical (non ab initio): Hückel1/Tight binding2, Hubbard, ...
• Wavefunction methods [Trygve Helgaker, Poul Jørgensen, Jeppe Olsen: Molecular Electronic‐Structure Theory. John Wiley and Sons, Ltd: 2000. DOI: 10.1002/9781119019572]

• Hartree method. Caution: mathematicians use this name for a related method.
• Hartree-Fock (HF): can be Restricted (RHF), Restricted Open (ROHF), Unrestricted (UHF) or Generalized (GHF).
• Configuration Interaction (CI), up to order $$N$$, which is equivalent to Full-CI1/exact diagonalization2/direct variational approach3.
• Multi-Configuration Self Consistent Field (MCSCF). [Chem. Rev. 2012, 112 (1), 108–181.]
• Complete Active Space (CASSCF).
• Two Configurations.
• Manual selection of configurations or other.
• Multi-Reference Configuration Interaction (MRCI)
• CASPT2, NEVPT2, partially (PC) and strongly contracted (SC) variants of the N-electron valence state second-order perturbation theory
• Multi-Reference Coupled Cluster methods (MRCC)
• Many Body Perturbation Theory methods (MBPT) [Isaiah Shavitt, Rodney J. Bartlett: Many-Body Methods in Chemistry and Physics, MBPT and Coupled-Cluster Theory. Cambridge University Press: Cambridge, 2009. DOI: 10.1017/CBO9780511596834]

• Coupled Cluster methods and its simplifications
• Möller-Plesset perturbation theory of order $$n$$ (MPn)
• Random Phase Approximation (RPA)
• GW-Approximation
• Explicit correlated methods R12/F12 [Chem. Rev. 2012, 112 (1), 75–107.]

• Valence Bond Theory [Sason Shaik, Philippe C. Hiberty: A Chemist's Guide to Valence Bond Theory. John Wiley & Sons, Inc.: 2008. DOI: 10.1002/9780470192597]
• Density Functional Theory (DFT). [Robert G. Parr, Yang Weitao: Density-Functional Theory of Atoms and Molecules. Oxford University Press: USA, 1994. ISBN: 9780195092769]

• Thomas-Fermi theory. Orbital-free methods.
• Kohn-Sham. Many density functional approximations developed, families are:
• LSDA
• GGA, hybrid GGA
• MGGA, hybrid MGGA
• OEP
• Pair density functional theory [Comput. Theor. Chem. 2013, 1003, 91–96.]
• Natural Orbital Functional Theory (NOFT) [S. Goedecker, C. J. Umrigar: Natural Orbital Functional Theory. In: J. Cioslowski (eds) Many-Electron Densities and Reduced Density Matrices. Mathematical and Computational Chemistry. Springer: 2000, Boston, MA. DOI: 10.1007/978-1-4615-4211-7_8]
• Geminal Functional Theory [J. Chem. Phys. 2000, 112 (23), 10125–10130.]
• k-particle density cummulants methods [Int. J. Quantum Chem. 2003, 95 (4-5), 404–423.]
• Density Matrices methods
• Dynamical Mean Field Theory (DMFT). [J. Chem. Phys. 2011, 134 (9), 094115., arXiv:1012.3609 [cond-mat.str-el]] Starts with a Hubbard model of a lattice problem, then connects it to an impurity problem that can be solved with other methods.

Non-deterministic (stochastic)

• Quantum Monte Carlo [Chem. Rev. 2012, 112 (1), 263–288.]
• Variational Monte Carlo (VMC)
• Fixed-Node Diffusion Monte Carlo (FNDMC).
• LR-DMC, lattice-regularized diffusion Monte Carlo
• Self-Healing Diffusion Monte Carlo
• Green Function Monte Carlo (GFMC). GF2, self-consistent second-order Green’s function
• Auxiliary Field Quantum Monte Carlo (AFQMC)
• Reptation Quantum Monte Carlo
• Full CI Quantum Monte Carlo (FCIQMC) [J. Chem. Phys. 2009, 131 (5), 054106.]
• BDMC, bold diagrammatic Monte Carlo
• Time-Dependent Quantum Monte Carlo

There are also mixtures of various methods. The methods above have been tested with molecules by the quantum chemistry community but there are others that have been proven only for one dimension or toy models. Operator methods are one example.

Many of them try to approximate the Full CI tensor that is described in a basis of determinant functions. It is impossible to treat such tensor for medium to large systems so the point is to take the minimum number of determinants that contribute significantly. The difference lies in how they choose those determinants. There are also other important distinctions among them, like if they are variational, perturbative or none, size-extensive, size-consistent, ...

In summary, each has its advantages and disadvantages. I recommend anyone interested to read SOLVING THE SCHRÖDINGER EQUATION. Has Everything Been Tried? 2011, Imperial College Press. Ed.: Paul Popelier.

Furthermore, the analytic properties (read regularity) of the wavefunction can be exploited to devise new methods. As Harry Yseretant puts it in his book:

Approximating the solutions is thus inordinately challenging, and it is conventionally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book tries to convince the reader that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons.

Also, I would like to note that there is a collaborative project under the hood of the Simons Foundation whose purpose is precisely finding better methods to solve the electronic structure problem. And, of course, there are many more people working on the same task.

Footnotes

1. Naming adopted by the chemistry community
2. Naming adopted by the physics community
3. Naming adopted by the mathematics community

References

1. Mario Motta, David M. Ceperley, Garnet Kin-Lic Chan, John A. Gomez, Emanuel Gull, Sheng Guo, Carlos A. Jiménez-Hoyos, Tran Nguyen Lan, Jia Li et al. Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods. Phys. Rev. X 2017, 7, 031059. DOI: 10.1103/PhysRevX.7.031059
• Just transforming all the citation made me realise how much work this was in the first place. Thanks for the effort. I have put up a bounty, which I will reward to your answer after the weekend. Sep 26 '18 at 13:55
• @Martin-マーチン Thanks to you for taking the initiative of doing the boring and hard job of making the references consistent with the style accepted in the site. Sep 26 '18 at 17:00
• You are welcome! And for future reference, I can recommend the user script which is presented in the following answer: Are there any tools to help me format citations appropriately? It makes the job much easier. Sep 26 '18 at 17:14