Expression for MP2 energy

Let us note first that, when calculating the mean value of the Hamiltonian in the standard Hartree-Fock method, we automatically obtain the sum of the zeroth order energies X /and the first-order correction to the energy ( RHF 77d) Rjjp)  

So what is left to be done (in the MP2 approach) is the addition of the second order correction to the energy (p. 208, the prime in the sununation symbol indicates that the term making the denominator equal to zero is omitted), where, as the complete set of functions, we assume the Slater determinants corresponding to the energy (they are generated by various spinorbital occupancies)  

In such a case, we take as the functions only doubly excited Slater determinants i/r, which means that we replace the occupied spinorbitals p,b q, 

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Another Unear-scaUng MP2 algorithm was proposed in [129] that is based on the atomic-orbitals Laplace-transform (LT) MP2 method [137]. In this method, the energy denominators (5.32) in (5.30) are ehminated by Laplace-transformation, which paves the way to express the MP2 energy directly in the basis. The price to pay is the additional Laplace integration, which is carried out by quadrature over a few (8-10) points. For each of the quadrature points an integral transformation has to... 

The selection rules for the QM harmonic oscillator pennit transitions only for An = 1 (see Section 14.5). As Eq. (9.47) indicates diat the energy separation between any two adjacent levels is always hm, the predicted frequency for die = 0 to n = 1 absorption (or indeed any allowed absorption) is simply v = o). So, in order to predict die stretching frequency within the harmonic oscillator equation, all diat is needed is the second derivative of the energy with respect to bond stretching computed at die equilibrium geometry, i.e., k. The importance of k has led to considerable effort to derive analytical expressions for second derivatives, and they are now available for HF, MP2, DFT, QCISD, CCSD, MCSCF and select other levels of theory, although they can be quite expensive at some of the more highly correlated levels of theoiy. 

P PjV 2 (pp4>q)) [ qj / ) reading/writing to disk, it is desirable also to have direct algorithms for electron correlation method. Direct in this context means that the integrals are calculated as needed and then discarded. The need for integrals over MOs instead of AOs, however, makes the development of direct methods in electron correlation somewhat more complicated than at the HF level. Consider for example the MP2 energy expression." ... 

It should be noticed that the authors did not provide an analytical expression for the second derivative of the total energy per unit cell in the MP2 case. 

Following the general theory of gradients in MP theory [117], differentiation of the MP2 energy given in Eq. (6.30) with respect to the components mf of mj yields. (Note that this expression is valid only for one-electron perturbations and perturbation-independent basis functions. The more general case requires additional terms. (See [117].))... 

In this way, we have obtained an expression for the dominating contribution to the high-(/ i/2) PW/m increments to the MP2 energy for the SAP defined by T in terms of integrals involving known functions only. Hence, we are now in a position to find for the PW/m expansion the leading AECs, i.e., the coefficient of the (/ -t-1/2) term with minimum k. 

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