Operators Hartree-Fock

Boys and Cook refer to these properties as primary properties because their electronic contributions can be obtained directly from the electronic wavefunction As a matter of interest, they also classified the electronic energy as a primary property. It can t be calculated as the expectation value of a sum of true one-electron operators, but the Hartree-Fock operator is sometimes written as a sum of pseudo one-electron operators, which include the average effects of the other electrons. 

Since the Hartree-Fock operator 7 is readily seen to be Hermitian, it is apparent from Eq. (26) that the CMO s necessarily form an orthonormal set. 

Since the Hartree-Fock wavefunction 0 belongs to the totally symmetric representation of the symmetry group of the molecule, it is readily seen that the density matrix of Eq. (10) is invariant under all symmetry operations of that group, and the same holds, therefore, for the Hartree-Fock operator 7. 

Here in eq. (38) "EpqfpQN a.pag is new Hartree-Fock operator for a new fermions (25), (26), operator Y,pQRsy>pQR a Oq 0s%] is a new fermion correlation operator and Escf is a new fermion Hartree-Fock energy. Our new basis set is obtained by diagonalizing the operator / from eq. (36). The new Fermi vacuum is renormalized Fermi vacuum and new fermions are renormalized electrons. The diagonalization of/ operator (36) leads Jo coupled perturbed Hartree-Fock (CPHF) equations [ 18-20]. Similarly operators br bt) corresponds to renormalized phonons. Using the quasiparticle canonical transformations (25-28) and the Wick theorem the V-E Hamiltonian takes the form... 

We now consider the PPP, CNDO, INDO, and MINDO two-electron semiempirical methods. These are all SCF methods which iteratively solve the Hartree-Fock-Roothaan equations (1.296) and (1.298) until self-consistent MOs are obtained. However, instead of the true Hartree-Fock operator (1.291), they use a Hartree-Fock operator in which the sum in (1.291) goes over only the valence MOs. Thus, besides the terms in (1.292), f/corc(l) m these methods also includes the potential energy of interaction of valence electron 1 with the field of the inner-shell electrons rather than attempting a direct calculation of this interaction, the integrals of //corc(/) are given by various semiempirical schemes that make use of experimental data furthermore, many of the electron repulsion integrals are neglected, so as to simplify the calculation. 

The sum in (1.291) is over the occupied spatial orbitals of the molecule. F is the Hartree-Fock operator, and is a one-electron operator, as indicated... 

In this case the operator P is equivalent to the Hartree-Fock operator for the total wavefunction [Pg.104]

The correlation energy, Econ, is defined as the difference between If exact, the experimentally determined ground state energy of a system, and Ess, the expectation value of the Hartree-Fock operator. 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

The Hartree-Fock operator must be defined only in terms of jr-type orbitals i.e., all the terms Ja and Kj are dropped in Eq. (5). In order to correct for this omission, the one-electron part of the operator has to be modified accordingly, writing... 

The evaluation of the matrix elements of the Hartree-Fock operator is usually carried out with a number of approximations. The diagonal, one-electron terms,... 

The matrix elements of the Hartree-Fock operator reduce then to... 

The Hartree-Fock operator fi(k) is generally expressed as a sum of the two terms,... 

This can be rewritten in terms of the Hartree-Fock operator (equations 17 and 18) as... 

This is the Hartree-Fock equation for atoms, and F is the Hartree Fock operator. Equation (6.53) can be written in matrix form,... 

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 ... 

However, the vacant Hartree-Fock molecular orbital (MO) obtained as a by-product of the ground-state calculations are of little use for describing the excited states of a molecule. This is due to the fact that the vacant Hartree-Fock MOs correspond to the motion of an excited electron in the potential field of all N electrons rather than of N - 1 electrons, as must be the case (Slater, 1963). Hunt and Goddard (HG) (1963) have proposed modifying the Hartree-Fock operator in such a way that it would be possible to describe the motion of an excited electron in the potential VN 1 ... 

For singlet excitations a = —1 andb = 2 for tritlet excitations a = —, b = 0 h, J, and K are, respectively, the one-electron, the Coulomb, and the exchange operators. They have the same meaning as in the usual Hartree-Fock operator. Here Ff1 3 is the N-electron operator with the hole in the ith occupied MO. Thus, Hunt and Goddard have replaced a single Hartree-Fock operator by a whole set of operators [Eq. (28)] differing in the position of the vacancy. The spectrum of each of these operators is an orthonormal set of MOs ... 

As it is the case for complicated problems where expansions into known functions are required, the final computational expressions are cast in matrix form. By forming the expectation value of the monoelectronic Hartree-Fock operator and by applying the variational procedure for the LCAO coefficients, C (k), the following system of equations, of size is obtained ... 

We restrict ourselves here to results that were obtained with the Mj ller-Plesset partitioning of the Hamiltonian, which means that the Hartree-Fock operator was extracted from the Hamiltonian as the "unperturbed" operator and the rest of the Hamiltonian was taken as the perturbation. The formula... 

We multiply the Schrodinger equation (5.75) on the left by the inverse of the Hartree—Fock operator to obtain... 

A semi-empirical calculation method for ionization potentials has been developed, using the fact that a Slater determinant is defined only up to a unitary transformation (see Sect. 4.4) the canonical molecular orbitals closed-shell system, can be replaced by equivalent orbitals eo, almost completely localized. 

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