Two-electron integrals antisymmetrized

The construction (ij kl) is called an antisymmetrized two electron integral in physicists notation. 

Figure 1-2. Schematic diagrammatic representation of the E correction (Brandow skeletons). The horizontal lines represent the denominators, while the vertical bar separates the monomers A and B. The two-electron integral corresponding to the dotted interaction line is a Coulomb integral. The dashed interaction lines represent antisymmetric two-electron integrals of the monomers. Diagram (a) is the intermolecular perturbation theory form of the MP5 contribution s, diagram (d) of qQ(/7), while (b) and (c) are combinations of 7s T and E (I)...

Remembering that for antisymmetrized two-electron integrals in Dirac s notation, pqWrs) = - pqWsr) = - ( pllrs) = qpWsr), we may re-index sums and combine terms where appropriate to obtain... 

In the above equations, hpv are the usual one-electron integrals while [juv Ao] and [juA vo] are the standard bare and antisymmetrized two-electron integrals, respectively. To derive these formulae, one has merely to substitute the second quantized form of the total Hamiltonian and apply the above rules for the density matrix elements. The analogy of Eq. (27) to the corresponding HF formula is obvious. 

Using Eq. (30), the definition of the antisymmetrized two-electron integral, and the anticommutation relation of fermion annihilation operators, Eq. (33) may be written... 

The numerator of the diagram contains products of the one-electron integrals (if any) and the antisymmetrized two-electron integrals. The indices of the one- and two-electron integrals are assigned according to the rule... 

The closed-shell Fock matrix is a sum of the core Fiamiltonian matrix h and the two-electron part that has a form of a produa of the first-order density matrix P and the supermatrix (tensor) of antisymmetrized two-electron integrals r (see Eq. [3])... 

Choose one of the antisymmetric two-electron integrals in Eq. (9.131) (say the second), and draw a line ( dashed or wavy) to represent this interaction. In a formal scattering process, the left end of this interaction line represents the point where electron 1 enters and leaves, and the right end where electron 2 enters and leaves [see Fig. 9.2]. 

Because two-electron integrals often appear in the following combination, we introduce a special symbol for an antisymmetrized two-electron integral... 

The summation of antisymmetrized two-electron integrals is thus over all unique pairs of spin orbitals Xm Xn occupied in C>. This observation suggests a simple mnemonic device for writing down the energy of any single determinant in terms of one- and two-electron integrals over spin orbitals. Each occupied spin orbital Xi contributes a term to the energy and... 

Recall that the antisymmetrized two-electron integral is defined as... 

Here, Sp are the orbital energies and ( pq rsy are the antisymmetrized two-electron integrals. As usual, indices denote occupied spin orbitals... 

Fock matrix elements [6] and of the antisymmetrized two-electron integrals, in the... 

Table 2 Efficient Reformulation of the CCSD Equations in Terms of Intermediates. As Usual, /, Denote.s in the Equatioas the Corresponding Fock Matrix Elements, (pq rs) Antisymmetrized Two-electron Integrals, and P.(pq) the Antisymmetrization Operator (P.(pq)Z - pq = Z(- pq ) — Z - qp. The Eiastein Summation Convention is Employed...

Since the MP perturbation operator involves a two-electron operator, the standard Slater rules for orthonormal spin orbitals can be applied when calculating Koi, V , etc. All matrix elements can be expressed in terms of double-bar integrals (ij ab), which are antisymmetrized two-electron integrals of the general type pq rs) ... 

Each doubly excited determinant gives a contribution to the MPl wave function that is [x-oportional to the antisymmetrized two-electron integral weighted by a denominator containing the associated orbital energies. 

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