The Choice of Zero-order Hamiltonian

4 The Choice of Zero-order Hamiltonian. - The utility of low-order 

The use of shifted denominators may be interpreted as the summation of certain diagonal terms in the higher-order Moller-Plesset expansion though infinite 

The choice of zero-order hamiltonian is, to some extent, arbitrary. By using a 

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The energy selected basis (ESB) presented here has the best of both worlds. The choice of zero order reduced Hamiltonians provides a yood zero order basis energy selection from this basis provides an extremely compact basis with limited spectral... 

However, this simple modification shows that there is some choice of zero-order hamiltonian which yields the exact correlation in second order. In such a case the remainder term is zero, 3R. 

The use of M0ller-Plesset or Hartree-Fock model to label particular choices of zero-order Hamiltonian in many-body perturbation theory dates from the work of Pople et al. [2] and of Wilson and Silver [3]. In their original publication of 1934, MpUer and Plesset [4] did not recognize the many-body character of the theory in the modern (post-Brueckner) sense. 

In the present section, we do not specify the form of the Hamiltonian operator. The results obtained here are therefore valid for any choice of zero-order operator and for any choice of perturbation operator. In the remaining sections of this chapter, we shall be less general and specify in detail the unperturbed and perturbed Hamiltonian operators. In particular, we shall concern ourselves with the application of perturbation theory to the calculation of the electron correlation energy, but it should be understood that the techniques presented here are valid also in other situations. For example, RSPT as presented here provides a convenient theoretical framework for the systematic study of time-independent molecular properties. 

Hamiltonian proposed by Muller and Plesset gives rise to a very successful and efficient method to treat electron correlation effects in systems that can be described by a single reference wave function. However, for a multireference wave function the Moller-Plesset division can no longer be made and an alternative choice of B(0> is needed. One such scheme is the Complete Active Space See-ond-Order Perturbation Theory (CASPT2) developed by Anderson and Roos [3, 4], We will briefly resume the most important definitions of the theory one is referred to the original articles for a more extensive description of the method. The reference wave function is a CASSCF wave function that accounts for the largest part of the non-dynamical electron correlation. The zeroth-order Hamiltonian is defined as follows and reduces to the Moller-Plesset operator in the limit of zero active orbitals ... 

For single reference perturbation theory, there is a choice of reference hamil-the Moller-Plesset and Epstein-Nesbet zero-order hamiltonians were two choices considered in the early literature (see, for example, Ref. 54). 

Hd) is called the perturbing Hamiltonian or the first-order Hamiltonian. The role of X is somewhat abstract. There may be no physical way of smoothly changing a system from some unperturbed model system to the system of interest however, the parameter X is a device that accomplishes this in a mathematical sense. When X is set to zero. Equation 8.70 gives the Hamiltonian for the zero-order or unperturbed situation, whereas when X is set to 1.0, the Hamiltonian of interest results. With this embedded parameter, the Schrodinger equation with H is really a family of equations covering all the choices of X values. Perturbation theory is a means of dealing with the entire family of equations, even though in the end, one may be interested only in the case where X = 1. 

This choice of ensures that, when 0) reduces to a single-determinant closed-shell state, the zero-order Hamiltonian (14.7.4) reduces to the Mpller-Plesset Fock operator. 

The zero-order induction approach differs from its parent R-SRS+ELHAV (Eq. 73) or R-SRS+SAM (Eq. 79) theory in the specific choice of the zero-order Hamiltonian and the regular part of the perturbation operator. These operators are replaced by new operators Hq = Ha + Hb and Vp defined such that the effects of the (regularized) induction interaction are included in the zeroth order. Specifically, one can set Ha = Ha + 2B and Hb = Hb + where... 

In this section we will discuss the problem of the nature of intramolecular interstate coupling and criteria for the choice of a basis set and consider the aspect whether these basis sets mentioned above is appropriate for describing the electronic relaxation processes. We begin by considering a complete set of zero-order functions, where electronic and nuclear motions have been separated arbitrarily. The index 0 refers to electronic state, while the second index v labels the vibrational state. Utilizing the completeness assumption )(av = 1 and the trivial relation I av) (va H o v ) (v a, we decompose the total Hamiltonian in the form... 

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