Hartree-Fock equations/theory closed-shell

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

It would, therefore, be pedantic to insist that the Hartree-Fock equations always be solved in their full generality when, in most cases, considerable savings can be effected by constraining the molecular orbitals to have certain molecular symmetries. Most routine uses of LCAOSCF theory can safely assume that the optimum MOs will have the symmetry of the molecular framework the dominant nuclear-attraction term in the Hamiltonian. The situation is similar to the use of the closed-shell constraint in place of a full GUHF solution for molecules of conventional known closed-shell structure. In any case, the numerical techniques used to impose spatial symmetry constraints on the MOs are very similar to techniques used for a variety of other purposes in LCAOSCF theory—the transformation of the Fock matrix to a new basis—so we include a discussion of the techniques here. 

The nonrelativistic Hartree-Fock theory (abbreviated HF in the equations to follow) formally resembles Dirac-Hartree-Fock (DHF) theory for the Dirac-Coulomb Hamiltonian. Of course, for large c of, say, 10 to 10 a.u. they even yield the same results. For this reason we shall make an explicit comparison of both in this section. The total energy for closed-shell atoms after integration over all angular and spin coordinates is in both cases given by... 

In passing from the spin-orbital equations to the closed-shell orbital equations in Section 6.2, a constraint was introduced it was assumed that all spin-orbitals were pure space-spin products of the form

kP and that these were occupied in pairs with a common orbital factor 4>k- In other words, we developed a restricted Hartree-Fock (RHF) theory. It is possible, however, that an optimized single-determinant wavefunction in which such restrictions were absent would yield a lower variational energy, the spin-orbitals being neither pure nor paired. In that... 

With only a few electrons outside a closed shell, the above equations may be solved iteratively in essentially the same manner as the Hartree-Fock equations. Usually an algebraic approximation is used in which every is expressed as a linear combination of basis functions as in Section 6.2 and this leads to matrix equations that are more suitable for computational purposes. The actual techniques of optimization are similar to those used in multiconfiguration versions of MO SCF theory (Chapter 8) the more rapidly convergent procedures require also second derivatives of the energy expression, but these may be obtained in the same way as the first derivatives. 

Derive the detailed expression for the orbital Hessian for the special case of a closed shell single determinant wave function. Compare with equation (4 53) to check the result. The equation can be used to construct a second order optimization scheme in Hartree-Fock theory. What are the advantages and disadvantages of such a scheme compared to the conventional first order methods ... 

Open-shell Pseudohamiltonians.—The majority of atoms do not have valence structures which can be represented by the fully closed-shell wavefunction of equation (14), and consequently ab initio pseudopotentials cannot be derived directly from the theory outlined above. Acceptable wavefunctions for such atoms require either more than one determinant or the use of the symmetry-equivalenced or generalized Hartree-Fock method, and usually include partially filled shells. The total all-electron wavefunction may be symbolically expressed in terms of four subspaces,... 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

For the development below, we will assume a closed-shell situation, with all electrons paired in molecular orbitals. In such a case OfSj = 1. In very many cases, however, an unrestricted Hartree-Fock (UHF) scheme is utilized for ground state properties. This theory is reasonably accurate for those cases in which each open-shell orbital has an electron of the same spin, i.e., the case that an open-shell has maximum multiplicity. In the UHF scheme Eq. [4] does not hold. Two Fock equations result, one for a and one for 3 spin molecular orbitals. In cases in which excited state properties are required, Eq. [4] is forced to hold in order to yield spectroscopic states, of known multiplicity. OfSJ can then become quite complex, and affects the form of the Fock operators that follow. ... 

The basic idea behind the application of DPT to the relativistic Hartree-Fock (HF) equations is simple. We start from the energy expression and the stationarity condition of relativistic closed-shell Hartree-Fock theory. 

The two fundamental building blocks of Hartree-Fock theory are the molecular orbital and its occupation number. In closed-shell systems each occupied molecular orbital carries two electrons, with opposite spin. The occupied orbitals themselves are only defined as an occupied one-electron subspace of the full space spanned by the eigenfunctions of the Fock operator. Transformations between them leave the total HF wave function invariant. Normally the orbitals are obtained in a delocalized form as the solutions to the HF equations. This formulation is the most relevant one in studies of spectroscopic properties of the molecule, that is, excitation and ionization. The invariance property, however, makes a transformation to locahzed orbitals possible. Such localized orbitals can be valuable for an analysis of the chemical bonds in the system. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

The Hartree-Fock method is the simplest ab initio approach. We can equate closed-shell HF theory to single determinant theory, and we are thus interested in finding a set of spin orbitals... 

An approach for treating open-shell molecular systems by a closed-shell formalism which utilizes the similarity between the SCF equations (see Density Functional Theory (DFT), Hartree Fock (HF), and the Self-consistent Field) and those for a fictitious closed-shell system in which the odd electron is replaced by two half-electrons. 

The theory outlined in the previous sections is formally applicable to arbitrary interacting closed-shell systems. However, it cannot be applied in practice to systems larger than two-electron monomers since the resulting perturbation theory equations are too difficult to. solve without some systematic approach to the many-electron problem. The obstacle encountered here is analogous to the standard electron correlation issue and its solution requires techniques quite similar to those of the conventional many-body perturbation theory (MBPT). One takes the product of monomer Hartree-Fock determinants... 

Another category of approaches that avoids the symmetry breaking problem of the orbitals for the target state is based on using a related, well-behaved HF reference instead. Examples of such techniques include quasi-restricted Hartree-Fock coupled-cluster (QRHF CC) (11,19), symmetry adapted cluster configuration interaction (SAC-CI) (22,23), coupled-cluster linear response theory (CCLRT) (24-26) or equivalently equation-of-motion coupled-cluster (EOM-CC) (27-32), Fock space multi-reference coupled-cluster (FSMRCC) (33-37), and similarity transformed equation-of-motion coupled-cluster (STEOM-CC) (38-40). In the case of NO3 and N03, the restricted Hartree-Fock (RHF) orbitals of the closed-shell N03 anion system can be used as the reference. The anion orbitals are stable with respect to symmetry perturbations, and the system does not suffer from the artifactual symmetry breaking of the neutral and cation. 

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