Hartree Coulomb operator

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 ... 

We introduce the change of the metric characteristic of DPT, and expand in powers of c. To 0(c ) we get the non-relativistic Hartree-Fock equations in Levy-Leblond form. The leading relativistic correction to the energy is then expressible in terms of nonrelativistic HF spin orbitals or rather the corresponding lower components xf - For the Dirac-Coulomb operator we get after some rearrangement [17, 18] ... 

The expression for the lowest order contribution to the parity violating potential within the Dirac Hartree-Fock framework is identical to that within the relativistically parameterised extended Hiickel approach in eq. (146). The difference is, however, that in DHF typically atomic basis sets with fixed radial functions are employed (see [161]) and that the molecular orbital coefficients are obtained in a self-consistent Dirac Hartree-Fock procedure. Computations of parity violating potentials along these lines have occasionally been called fully relativistic, although this term is rather unfortunate. In the four-component Dirac Hartree-Fock calculations by Quiney, Skaane and Grant [155] as well as in those by Schwerdtfeger, Laerdahl and coworkers [65,156,162,163] the Dirac-Coulomb operator has been employed, which is for systems with n electrons given by... 

In fact, this is the principle role of the exchange term to cancel the unphysical self-repulsion in the Coulomb sum. It is the difference between the Hartree and the Hartree-Fock methods, and the reason why all the MOs are the eigenfunctions of the same Hartree-Fock operator, while a separate Hartree operator is needed for each MO which excludes the self-repulsion for that MO. 

The difference between this Fock operator and the Hartree-Fock counterpart in Eq. (2.51) is only the exchange-correlation potential functional, Exc, which substitutes for the exchange operator in the Hartree-Eock operator. That is, in the electron-electron interaction potential, only the exchange operator is replaced with the approximate potential density functionals of the exchange interactions and electron correlations, while the remaining Coulomb operator, Jj, which is represented as the interaction of electron densities, is used as is. The point is that the electron correlations, which are incorporated as the interactions between electron configurations in wavefunction theories (see Sect. 3.3), are simply included... 

These equations are quite analogous to those for spin orbitals, except for the factor of 2 occurring with the coulomb operator. The sum in (3.1 ) is, of course, over the N/2 occupied orbitals The closed-shell spatial Hartree-Fock equation is just... 

In 2003, Heyd et al. proposed to use a screened Coulomb operator in the Hartree-Fock exchange part of hybrid functionals [41]. The Coulomb operator was split into short-range (SR) and long-range (LR) components. 

We are immediately confronted with the problem of how to find the unknown exchange-correlation energy Exc, which is replaced also by an unknown exchange-correlation potential in the form of a functional derivative Vxc = We obtain the Kohn-Sham equation (resembling the Fock equation) -jA -F ng (/>i = eifi, where wonder-potential t)g = t) -F Vcoyi -F i xc, fcoul stands for the sum of the usual Coulombic operators (as in the Hartree-Fock method) (built from the Kohn-Sham spinorbitals) and vxc is the potential to be found. 

In the Hartree-Fock operator F the term Hq is the one-electron operator (4.8). The action on the function v <(r) of the Coulomb operator J and exchange operator K is determined in the following way. Denote by p r,r ) the mixed electron density with fixed spin (spinless electron-density matrix)... 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

A special case of this approach is represented by the Hartree-Fock equations, where the effective operator Heff contains the usual kinetic (T), nuclear attraction (U), Coulomb (J), and exchange (K) components such that... 

The sum over coulomb and exchange interactions in the Fock operator runs only over those spin-orbitals that are occupied in the trial VF. Because a unitary transformation among the orbitals that appear in F leaves the determinant unchanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to choose such a unitary transformation to make the Ey matrix diagonal. Upon so doing, one is left with the so-called canonical Hartree-Fock equations ... 

Jj — K,j operating on / gives the expression in Hartree-Fock (HF) theory for the effective Coulombic repulsion energy between an electron /,- and the pair of... 

Whereas Si and s2 are true one-electron spin operators, Ky is the exchange integral of electrons and in one-electron states i and j (independent particle picture of Hartree-Fock theory assumed). It should be stressed here that in the original work by Van Vleck (80) in 1932 the integral was denoted as Jy but as it is an exchange integral we write it as Ky in order to be in accordance with the notation in quantum chemistry, where Jy denotes a Coulomb integral. 

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