The J and K Operators

We saw earlier that the variational energy for a closed-shell state formed from electron configurations such as 

The sums run over the occupied orbitals note that we have not made any reference to the LCAO approximation. The energy expression is correct for a determinantal wavefunction irrespective of whether the orbitals are of LCAO form or not. 

It is sometimes useful to recast the equation as the expectation value of a sum of one-electron and pseudo one-electron operators 

The operator h is a one-electron operator, representing the kinetic energy of an electron and the nuclear attraction. The operators J and K are called the Coulomb and exchange operators. They can be defined through their expectation values as follows. 

We will meet these two operators again when we study density functional theory in Chapter 13. 

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The HF equations must be solved iteratively beoause the J- and K. operators in F depend on the orbitals ( ). for whioh solutions are sought. Typioal iterative sohemes begin with a guess for those ([). that appear in T", whioh then allows f to be fonned. Solutions to = e.. are then found, and those (j). whioh possess the spaoe and... 

The Fock operator is an effective one-electron energy operator, describing the kinetic energy of an electron, the attraction to all the nuclei and the repulsion to all the other electrons (via the J and K operators). Note that the Fock operator is associated with the variation of the total energy, not the energy itself. The Hamilton operator (3.23) is not a sum of Fock operators. 

Since the J and K operators depend on the occupied orbitals, the pseudoeigenvalue problem must be solved iteratively until consistency is achieved between orbitals that determine J and K and those that emerge as eigenfunctions of Heff, which in this approximation is known as the Fock operator F. 

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. 

Dirac Notation for Integrals Energy Functional to Be Minimized Energy Minimization with Constraints Slater Determinant Subject to a Unitary Transformation The J and K Operators Are Invariant Diagonalization of the Lagrange Multipliers... 

As in Gaussian 94, PS methods utilize the Roothaan-Hall equations with a Gaussian atomic basis set. However, a numerical grid is used to facilitate the construction of the J and K operators. The basic equations of the method " are given by sums over a mesh of grid points (g) ... 

Here, ma is the mass of the nucleus a, Zae2 is its charge, and Va2 is the Laplacian with respect to the three cartesian coordinates of this nucleus (this operator Va2 is given in spherical polar coordinates in Appendix A) rj a is the distance between the jth electron and the a1 1 nucleus, rj k is the distance between the j and k electrons, me is the electron s mass, and Ra>b is the distance from nucleus a to nucleus b. 

By means of the few basic ideas exposed so far it is possible to follow a COSY experiment. Say the two coupled nuclei are I and AT, the coupling constant between the two being J. We can follow the evolution of one spin the evolution of the other can be easily obtained by swapping the 7 and K operators. After the first 90°j, pulse, z magnetization is converted into x magnetization. Then, evolution during t follows... 

Direct Cl procedures of a quite general kind (28) may be formulated as a matter of constructing large numbers of J and K operators, usually frcm a transformed integral list/ and hence directly in the m.o. basis Algorithms as described above are also applicable to this case, and should proceed at 135 Mflops given a reasonably large external space. 

Derive equations analogous to those that lead up to (6.2.20), for a closed-shell state, starting from the spinless energy expression (S.3.18). Hence justify the conclusions stated on p. 171. [Hint Introduce spinless J and K operators and verify the alternative form (6.2.24). Insert finite basis approximations in the 1- and 2-electron integrals and pass to the equivalent matrix forms (p. 171).]... 

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

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,. 

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