Operators core Fock

The key process in the HF ab initio calculation of energies and wavefunctions is calculation of the Fock matrix, i.e. of the matrix elements Frs (Section 5.2.3.6.2). Equation (5.63) expresses these in terms of the basis functions and the operators //core, J and K, but the J and K operators (Eqs. 5.28 and 5.31) are themselves functions of the MO s i// and therefore of the c s and the basis functions Fock matrix to be efficiently calculated from the coefficients and the basis functions without explicitly evaluating the operators J and K after each iteration. This formulation of the Fock matrix will now be explained. 

This further simplifies to give an effective Fock operator, core n o (21)... 

The simplest way to deal with energetically low-lying closed-shell core orbitals is to take them directly from a preceding SCF calculation without further optimization. In this case one has to eliminate all rows and columns corresponding to core orbitals from the matrices P, Q, R, A, B, etc., and replace the one-electron Hamiltonian h by a core Fock operator F. This operator is calculated in the AO basis according to... 

In the equation above, is the core Fock operator, and sum over runs over all the doubly occupied active orbitals of cf>. F is the two-body portion of H in normal order with respect to We do not explicitly indicate here and later which < 0/u, been used as the vacuum, since it would be clear from the functions the operators act upon. 

The sums now range over the valence electrons. The one-electron operator has been replaced by the core Fock operator. 

If the core orthogonality is retained, there is no necessity to insert the projection operators around the core Fock operator in (20.5), but we must still ensure that core spinors are kept out of the valence space. In an atom it is easy to maintain the orthogonality, but in a molecule the basis functions on another center expand into a linear combination of functions on the frozen-core center, including core spinors. 

To construct the Fock matrix, eq. (3.51), integrals over all pairs of basis functions and the one-electron operator h are needed. For M basis functions there are of the order of of such one-electron integrals. These one-integrals are also known as core integrals, they describe the interaction of an electron with the whole frame of bare nuclei. The second part of the Fock matrix involves integrals over four basis functions and the g two-electron operator. There are of the order of of these two-electron integrals. In conventional HF methods the two-electron integrals are calculated and saved before the... 

In the Fock operator, the core Hamiltonian h( 1) does not depend on the orbitals, but the Coulomb and exchange operators (1) and ( 1) depend on ( 1). If (1,2,3,..., Ne) is constructed from the lowest energy Ne orbitals, one has the lowest possible total electronic energy. By Koopmans theorem, the negative of the orbital energy is equal to one of the ionization potentials of the molecule or atom. 

The ECP method dates back to 1960, when Phillips and Kleinman suggested an approximation scheme for discarding core orbitals in band calculations [1]. They replaced the full Fock-operator with the following operator ... 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

The. totally symmetric operators M(r) are of the same form as the operators fi(r). The c s are atomic core orbitals expressed in the full all-electron basis set, and Fval are normal Fock operators defined in the valence space only. The valence orbitals v = cpXv are expressed in an appropriate valence basis set, determined through some optimization procedure, which is considerably smaller than the original all electron basis set. 

Bonifacic and Huzinaga[3] use explicit core orbital projection operators, while orbital angular momentum projection operators are used by Goddard, Kahn and Melius[4], by Barthelat and Durand[5] and others. Explicit core orbital projection operators can, in the full basis set, be viewed as shift operators which ensure that the first root in the Fock matrix really corresponds to a valence orbital. However, in applications the basis set is always modified and the role of the core orbital projection operators thus partly changes. 

Clearly any attempt to base FeK on such molecularly defined cores defeats the aims of pseudopotential theory. However, the approximate invariance of atomic cores to molecule formation implies that, of the total of Na electrons which could be associated with the centre A in an atomic calculation, nx are core electrons and n K will contribute to the molecular valence set. Thus we can define a one-centred Fock operator ... 

This Fock operator has been derived starting from the assumption of a Hartree-Fock valence function valence electrons has little influence on the core electrons, so that the many-electron valence hamiltonian may be similarly approximated as... 

The Fock operator defined in eq. (2.7) above, here rewritten for a closed shell system, can be diagonalized and expressed in terms of the core (c) valence (v) and excited state (e) projectors as7 ... 

As can be seen from Equation 6.33, the one-electron part of the Fock operator (cf. Equation 6.25 and Equation 6.26) is set to be proportional to the overlap between atoms A and B with respect to the two specified AOs. Note that this approach violates the NDDO formalism consequently, the approach was called MNDO, and the successor models AMI, PM3, and PM5 are in fact MNDO-type methods. The one-electron matrix element of Equation 6.33 represents the kinetic and potential energy of the electron in the field of the two nuclei (represented by their core charges), and the P terms are the respective single-atom resonance integrals. 

The form of the wave operators need not be defined, but, in principle, they can describe any type of wavefunction, for example, Hartree-Fock or coupled-cluster wavefunctions. However, at their core, they always consist of strings of creation operators. We define the supermolecular wavefunction as... 

The problem is now solved again by an iterative process, which starts with a choice of the x set and the expansion (6.58). The Hartree-Fock operator F and the matrix representation Fx are calculated, (6.64) is solved for the orbital energies, and these are used to compute a new set of coefficients in (6.63). If these are different from the starting set, the cycle is repeated until the self-consistent-field limit is reached. The total electronic energy is obtained by adding the SCF energy to the core energy for the lowest occupied n/2 levels ... 

Once a nodeless orbital has been generated the one-electron atomic Fock equation is easily inverted to produce a (radially) local operator, the EP, which represents the core-valence interactions (22,23). 

The method discussed here for the inclusion of relativistic effects in molecular electronic structure calculations is grounded in the Dirac-Fock approximation for atomic wave functions (29). The premise is that the major relativistic effects of the Dirac Hamiltonian are manifested in the core region, involving the core electrons, and that these effects propagate to the valence electrons. In addition, there are direct relativistic effects on valence electrons penetrating into the core region. Insofar as this is true, the valence electrons can be treated using a nonrelativistic Hamiltonian to which is added an operator, the relativistic effective core potential (REP). The REP formally, incorporates relativistic effects due to core electrons and to interactions of valence electrons with core electrons in an internally consistent way. 

Fig. 6. Average relativistic effective core potential and relativistic effective core potential energy curves for two states of Bi2. HF, Hartree-Fock GVB(pp), eight-configuration perfect-pairing generalized valence bond FVCI, full-valence Cl based on the GVB(pp) wave functions FV7R, full-valence Cl plus single and double promotions to virtual MOs relative to seven-dominant configurations. (The FVCI and FV7R calculations include the REP-based spin-orbit operator.)...

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