Closed-shell case

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

Bacskay G B 1981 A quadratically convergent Hartree-Fock (QC-SCF) method. Applications to the closed-shell case Chem. Phys. 61 385... 

We then allow Ri and R2 to vary, subject to orthonormality, just as in the closed-shell case. Just as in the closed-shell case, Roothaan (1960) showed how to write a Hamiltonian matrix whose eigenvectors give the columns U] and U2 above. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

You probably noted that the original papers were couched in terms of HF-LCAO theory. From Chapter 6, the defining equation for a Hamiltonian matrix element (in the usual doubly occupied molecular orbital, closed-shell case) is... 

Besides these many cluster studies, it is currently not knovm at what approximate cluster size the metallic state is reached, or when the transition occurs to solid-statelike properties. As an example. Figure 4.17 shows the dependence of the ionization potential and electron affinity on the cluster size for the Group 11 metals. We see a typical odd-even oscillation for the open/closed shell cases. Note that the work-function for Au is still 2 eV below the ionization potential of AU24. Another interesting fact is that the Au ionization potentials are about 2 eV higher than the corresponding CUn and Ag values up to the bulk, which has been shown to be a relativistic effect [334]. A similar situation is found for the Group 11 cluster electron affinities [334]. 

Finally, for the single-bonded closed-shell case the n monovalent ligands contribute n electrons for oml bonding, and the remaining k — n+ 2d electrons must... 

The XC energy represents the correction to the Coulomb energy for the self-energy of an electron in a many-electron system. The latter is due to both the direct self-energy of the electron as well as the redistribution of electronic density around each electron because of the Pauli exclusion principle and the Coulomb interaction. As an example, we now discuss the case of Fermi hole and the exchange energy in Hartree-Fock (HF) theory [16]. For brevity, we restrict ourselves to closed-shell cases. 

We limit ourselves to the closed-shell case. We try to solve this system iteratively, inserting n, = 1 and ria = 0 and neglecting X2 on the rhs in the first iteration. Then we obtain in the first iteration... 

Note that the sum over i in Y K and YgKex is a sum over spin orbitals. In addition, show that this Fock matrix can be further reduced for the closed shell case to ... 

Gdanitz and Ahlrichs devised a simpler variant of CPF, the averaged coupled-pair functional (ACPF) approach [30]. This produces results very similar to CPF for well-behaved closed-shell cases and is completely invariant to a unitary transformation on the occupied MOs. Its big advantage is that it can be cast in a multireference form. Multireference ACPF is probably the most sophisticated treatment of the correlation problem currently available that can be applied fairly widely, although it can encounter difficulties with the selection of reference spaces, as discussed elsewhere. 

B. D. Dunietz, R. B. Murphy, and R. A. Friesner,/. Chem. Phys., 110,1921 (1999). Calculation of Atomization Energies by a Multiconfigurational Localized Perturbation Theory Application for Closed-Shell Cases. 

In a symbolic notation, the Slater determinant for the electronic ground-state configuration, O,, as function of n electrons can be written in the closed-shell case as ... 

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... 

ZINDO57-58 is a semiempirical intermediate neglect of differential overlap/spectroscopy (INDO/S) based routine. It can be combined with an SOS method to calculate second-order nonlinear optical coefficients. ZINDO is parametrized to accommodate transition-metal calculations and is therefore suited for calculation on organometallic compounds. To achieve computational efficiency, some of the terms in Eq. (2) are replaced by empirical data or neglected. To see how the INDO/S does this, the closed-shell case will be examined.57 58 It is useful to introduce the following ... 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

The coefficients Cpt are obtained by minimizing the expectation value of the Hamiltonian for the wavefunction V from equation (2). In the more commonly occurring closed-shell case this gives the secular equation... 

We turn now to the problem of optimizing the non-linear parameters in a wavefunction. As mentioned in the introduction, for non-linear parameters (such as orbital exponents or nuclear positions) traditionally, non-derivative methods of optimization are used. However, if we wish to use a gradient method, for example, we must be able to obtain the required derivatives, subject to the constraints on the non-linear parameters and also subject to the condition that the constraints on the linear parameters continue to be bound during the variant of the non-linear parameters. In the usual closed-shell case, Fletcher5 showed how the linear constraint restriction could be incorporated, providing that one started from a minimum in the linear parameters. Assuming for the moment no particular constraints on the non-linear variables, then starting from a linear-minimum it is easy to see that... 

However, in a quantum chemical context there is often one overwhelming difficulty that is common to both Newton-like and variable-metric methods, and that is the difficulty of storing the hessian or an approximation to its inverse. This problem is not so acute if one is using such a method in optimizing orbital exponents or internuclear distances, but in optimizing linear coefficients in LCAO type calculations it can soon become impossible. In modern calculations a basis of say fifty AOs to construct ten occupied molecular spin-orbitals would be considered a modest size, and that would, even in a closed-shell case, give one a hessian of side 500. In a Newton-like method the problem of inverting a matrix of such a size is a considerable... 

Note that i and j run over the orbitals in this closed shell case. There are three different types of integral in equation (6.39), and they are defined in the manner shown below. 

Self-Consistent-Field s The Closed Shell Case. ttiere are presently two favoured methods for computing a 2-electron interaction matrix, G = 2J-K ... 

As a final variant the SCF procedure may be solved by a Newton Raphson technique, a very important component of which comprises a partial or complete 4-index tramsformation of integrals at each cycle. As we show below, the integral transformation procedure is highly vectorisable. We feel that such a technique will perhaps prove profitable in slowly convergent close shell cases or complicated open shell cases. 

Self-Consistent-Field The Open Shell SCF Case. The major point here is that the advantage of the Supermatrix methods is approximately halved over the closed shell case simply because one requires J and K matrices to be computed individually, necessitating the construction and use of 2 Supermatrices. Our present code based on the integral driven algorithm performs at 10.3 Mflops when processing cases described by Roothaan (22). 

We note that the basis orbitals k(r) are common for both a and / spin electrons, therefore, the evaluation of one- and two-electron integrals has to be performed only once (or they can be taken over from the corresponding closed-shell calculations). The procedure analogous to the one applied for the closed-shell case leads to a set of coupled complex pseudo-eigenvalue equations of the form... 

The reduction coefficients q and Q are independent of both the Cl coefficients Ajf and the orbitals, and are determined by the form of the many-electron wavefunction. For simple wavefunctions they are very simple e.g. in the closed-shell case, the configuration subscripts can be omitted, and = 2dij, — 23ii,3j,. It is useful to introduce the orbital density matrices yij and rij i as the weighted sum of the reduction coefficients... 

For open-shell and small (e.g. two-configuration) MCSCF wavefunctions, the construction of the AO density matrix according to Eq. (32) is computationally negligible compared to the evaluation of the integral derivatives. The open-shell case is thus computationally identical to the closed-shell case. 

Analytical second derivatives for closed-shell (or unrestricted Hartree-Fock (UHF)) SCF wavefunctions are used routinely now. The extension to the MCSCF case is relatively new, however. In contrast to the first derivatives, the coupled perturbed SCF equations have to be solved in order to calculate the second and third energy derivatives. The closed-shell case is relatively straightforward, and will be discussed. The multiconfigurational formalism is... 

The unrestricted Hartree-Fock (UHF) case is completely analogous to the closed-shell one. New terms do appear in the open-shell SCF and the few-configuration case. Nevertheless, the preferred technique is quite similar to the closed-shell case. In particular, the two-particle density matrix can be constructed from compact matrices, and the solution of the derivative Cl equations is very simple, due to the small dimension. 

The third derivative formula for the closed-shell case is fairly simple (Moccia, 1970 Pulay, 1983a) ... 

In this expression the subscript 0 denotes the zeroth-order function d>A >B where < > and d>B are SCF wavefunctions (single Slater determinants in the closed-shell case) of the respective monomers . The index p runs over all single and double excitations from the zero-order function. The same expression is also used by Magnasco and Musso ° but in this case index p runs over singly excited configurations only dispersion effects are not included in their treatment. In addition the unoccupied orbitals are expressed in the form of antibonding orbitals. 

When fhe lasf fwo indices are expanded from lower triangles to squares, symmetric and antisymmetric matrices are formed. The superscripts S and A denote these cases. Diagonal elements for the symmetric case have a different normalization factor. The rank of the spin-adapted H matrix for the closed-shell case is o- -v- -vo - -ov. ... 

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