What are the types of SCF?

Question

Many of us know the most common types of SCF
(though we can do better than Wikipedia at explaining them):

• RHF (Restricted Hartree-Fock), RKS (Restricted Kohn-Sham) [link to answer]
• UHF (Unrestricted Hartree-Fock), UKS (Unrestricted Kohn-Sham) [link to answer]
• ROHF (Restricted Open-Shell Hartree-Fock), ROKS (Restricted open-shell KS)

But there's also less commonly known (and more powerful!) single-reference SCF methods:

• GHF (Generalized Hartree-Fock), GKS (Generalized Kohn-Sham) [link to answer]
• PHF (Projected Hartree-Fock)
• DHF (Dirac-Hartree-Fock) or Dirac-Fock [link to answer]
• KR-DHF (Kramers Restricted DHF)
• KU-DHF (Kramers Unrestricted DHF)
• SOSCF (Second-order Hartree-Fock)
• TD-SCF (Time-dependent SCF)
• Complex GHF
• GHF (Generalized HF of Valatin in 1961).
• GVB (Generalized Valence Bond) [link to answer]

Let's be the resource where people can learn these methods succintly!
(3 or fewer paragraphs please).

• Multi-configurational SCF methods can instead go here.
• Note that RHF sometimes stands for Roothan-Hartree-Fock.

Answer 1

DHF: Dirac-Hartree-Fock (or "Dirack-Fock")

the DHF (Dirac-Hartree-Fock) or Dirac-Fock is the SCF method based upon four-component spinors (simply four-spinors), because of the four-component Dirac-Coulomb(-Breit/Gaunt) Hamiltonian. The 4-spinors decribe both positive - electronic - solutions as well as negative, or "positronic" solutions. We are interested in electronic solutions, which are considered for correlated calculations.

4-spinors can be either Kramers restricted (KR) or Kramers unrestricted (KU). Four-component calculations are demanding because of extra small-component (S) basis functions. Good approximations to the Dirac Hamiltonian are two-component (2c) methods. 2c SCF then becomes identical with GHF.

Program packages of my choice:

• DIRAC based upon 4c/2c KR spinors (has both DFT and ab initio methods)
• ReSpect upon KU spinors (has effective HF/DFT implementations)

Answer 2

GHF: Generalized Hartree Fock

In Restricted Hartree-Fock (RHF), the molecular orbitals are constructed as pairs, with a single spacial function being used to describe both an $\alpha$ and $\beta$ spin electron. Unrestricted Hartree-Fock (UHF) lifts this requirement, forming a unique set of MOs for $\alpha$ and $\beta$ spin. Generalized Hartree-Fock takes this one step further and allows each MO to be an arbitrary linear combination of $\alpha$ and $\beta$ spin. This makes the MOs expressible as spinors. RHF and UHF solutions can be seen as just special cases/subspaces of GHF solutions, which shows that the global minimum determined by GHF must be less than or equal to the global minimum of RHF/UHF.

GHF ensures that the system wavefunction will be a continuous function of nuclear position, which is useful in the context of ab initio molecular dynamics or modeling bond breaking. GHF also has a tendency to produce a unique global minimum, making wavefunction optimizations less complicated. One drawback of GHF solutions is that the true wavefunction should be an eigenfunction of $S_z$ or $S^2$, but this is not generally true for GHF [1]. GHF also has applications in studying magnetic phenomena, as it allows you to represent spin configurations that are not possible with RHF or UHF (e.g. noncollinear spins) [2].

Implementations:

• PySCF (also allows GKS: Generalized Kohn-Sham)
• Gaussian (at least as an internal option, IOp(3/116=7, 15, or 19) for real, complex, or spinor basis)
• QChem
• Chronus Quantum

Bibliography:

1. Sharon Hammes-Schiffer and Hans C. Andersen J. Chem. Phys., Vol. 99, No. 3, 1 August 1993
2. Feizhi Ding, Joshua J. Goings, Michael J. Frisch, and Xiaosong Li J.Chem.Phys. 141, 214111 (2014)

Answer 3

RHF: Restricted Hartree-Fock / RKS: Restricted Kohn-Sham

Restricted Hartree-Fock (RHF) is a self-consistent field approach: a mean-field approximation to the electronic, non-relativistic Schrödinger equation. The electron-electon interaction is modeled as a field influencing all electrons in the system, which are otherwise independent particles. Their "movement" and "positions" are described by orbitals, which are one-electron wavefunctions (WF). They are collected in a determinant (called Slater determinant in the case of HF) forming the many-electron WF, from which expectation values like energy or electron density can be calculated.

In RHF, a pair of two electrons of opposite spin is restricted to use the same spatial orbital. (Formally, there are also spin components, which together with the spatial orbital form the spin orbital.) All electrons are part of one such pair. In essence, all orbitals are always doubly occupied - at any geometry etc.

Restriced Kohn-Sham DFT uses the same basic approach, the technical difference boils down to a different potential describing to non-Coulombic part of the electron-electron interaction. Note that the determinant is now called the Kohn-Sham determinant and refers to a fictitious system used "only" to determine the kinetic energy component of the underlying equations. However, it can also be used to calculate expectation values.

One of the advantages of RHF/RKS is that the resulting many-electron WF is an eigenfunction of the total spin operator. Another advantage is the comparatively low computational cost and ease of implementation. For many ground-state compounds (think classical main-group organic chemistry) and their properties, RHF/RKS can be appropriately used. The disadvantage is that RHF/RKS do not work well when bonds are stretched or the state is not a closed-shell singlet. Similarly, most transition states are badly described. Post-RHF treatments can repair these deficiencies, albeit at large computational expense, with other approaches often being more cost-effective.

Answer 4

UHF: Unrestricted Hartree-Fock / UKS: Unrestricted Kohn-Sham

In UHF, the restriction for electrons to pair-wise share a spatial orbital is lifted. The spatial orbitals of spin-up and spin-down electrons can differ entirely, as is the case for the $\ce{H2}$ dissociation (which is described far better by UHF than RHF, though not quantitatively correct). Another application are simpler open-shell cases, such as the lithium atom.

The computational cost is a bit larger than for RHF because essentially all quantities appear twice (except for integrals over atomic orbitals) and are linked only through the Coulomb interaction between electrons of opposite spin. For closed-shell singlets, UHF is a waste of computational time, because at geometries close enough to the minimum, the UHF solution will collapse onto the RHF solution.$^1$

UHF has the distinct disadvantage that it generally does not yield pure spin states. Instead of a doublet, one would obtain a mixture of a doublet (usually dominant), a quartet, a sextet etc. This can be quantified as the "spin contamination". This can also make post-UHF treatments harder to implement.

Concerning UKS, we note that some functionals also consider the spin-density for energy determination with "local spin-density approximation" (LSDA) being the simplest form.

$^1$ SCF stability analysis can be applied to force a UHF solution not found by the convergence algorithm/initial guess combination: R Seeger, JA Pople, J Chem Phys 66, 3045 (1977), doi: 10.1063/1.434318

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A Szabo, NS Ostlund: Modern Quantum Chemistry, Dover Publications, 1996.

Answer 5

GVB: Generalized Valence Bond

Unlike the Hartree−Fock (HF) wave function, Generalized Valence Bond GVB accounts for nondynamical correlation in valence orbitals. Compared with the complete-active-space self-consistent field (CASSCF) wave function, obtaining the GVB wave function is in principle less computationally demanding since the number of configurations is smaller. Unfortunately, GVB becomes problematic for molecules larger than a few atoms since the number of linearly independent spin functions (which grows factorially) becomes prohibitively high, and there is no unique way of truncating that number.

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Ref: J. Chem. Theory Comput. 2019, 15, 8, 4430–4439 'Generalized Valence Bond Perfect-Pairing Made Versatile Through Electron-Pairs Embedding'