Perturbation theory zeroth order Hamiltonian

Definition of the Zeroth Order Hamiltonian in Multiconfigurational Perturbation Theory (CASPT2). 

E. Rosta and P. R. Surjan, Two-body zeroth order Hamiltonians in multireference perturbation theory the APSG reference state. J. Chem. Phys. 116(3), 878-890 (2002). 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

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 ... 

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

The effects of the off-diagonal terms when folded-in by perturbation theory are of two types. They can either produce operators of the same form as those which already exist in the Hamiltonian constructed from the Azl = 0 matrix elements (the zeroth-order Hamiltonian), or they can have a completely novel form. A good example of the former type is the second-order contribution to the rotational constant which arises from admixture of excited and A states,... 

Andersson, K. Different forms of the zeroth-order Hamiltonian in second-order perturbation theory with a complete active space self-consistent field reference function, Theor. Chim. Acta 1995, 91, 31-46. 

The CASSCF wavefiinction is used as reference function in a second-order estimate of the remaining dynamical correlation effects. All valence electrons were correlated in this step and also the 3s and 3p shells on copper. Relativistic corrections (the Darwin and mass-velocity terms) were added to all CASPT2 energies. They were obtained at the CASSCF level using first-order perturbation theory. A level-shift (typically 0.3 Hartree) was added to the zeroth order Hamiltonian in order to remove intruder states [30]. Transition moments were conputed with the CAS state-interaction method [31] at the CASSCF level. They were... 

This problem can be avoided, however, if an appropriate open-shell perturbation theory is defined such that the zeroth-order Hamiltonian is diagonal in the truly spin-restricted molecular orbital basis. The Z-averaged perturbation theory (ZAPT) defined by Lee and Jayatilaka fulfills this requirement. ZAPT takes advantage of the symmetric spin orbital basis. For each doubly occupied spatial orbital and each unoccupied spatial orbital, the usual a and P spin functions are used, but for the singly occupied orbitals, new spin functions. 

The space Hq is usually chosen by selecting d eigenfunctions of a zeroth order Hamiltonian and k, I and h are obtained using perturbation theory or one of its formally exact reformulations, for example, an iterative scheme. In general, only these operators are perturbatively expanded and not the model or the true eigenfunctions. By multiplying (2.2) on the left by / and (2.5) on the left by k, it follows that / and k are related to one another by [37]... 

M0ller and Plesset proposed an alternative way to tackle the problem of electron correlation [Moller and Plesset 1934] Their method is based upon Rayleigh-Schrodinger perturbation theory, in which the true Hamiltonian operator X is expressed as the sum of a zeroth-order Hamiltonian Xq (for which a set of molecular orbitals can be obtained) and a perturbation, f " ... 

Some of the more important aspects of the theory behind the method are described in the review. In particular, the choice of the zeroth-order Hamiltonian is discussed together with the intruder-state problem and its solution. A generalization of the method to a multistate perturbation approach is suggested. Problems specifically related to spectroscopic applications are discussed, such as the choice of the active space and the treatment of solvent effects. 

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

To begin, let us see what all the several forms of Canonical Perturbation Theories (CPT) provide. All the CPTs [45-53], including normal form theories [54,55], require that an M-dimensional Hamiltonian H(p, q) in question be expandable as a series in powers of s, where the zeroth-order Hamiltonian is integrable as a function of the action variables J only... 

The most popular way of including dynamic correlation upon a CASSCF reference wave function [54-57] is the second-order perturbation theory (CASPT2) developed by Roos and coworkers [58]. However, in contrast to the single-configurational case, where the definition of the zeroth-order Hamiltonian is universal and taken as the sum of the one-electron Fock operators, the generalization of the zeroth-order Hamiltonian to the multiconfigurational case is not straightforward [59, 60]. A different, theoretically more justified approach is to... 

When H is additively separable as in an independent-particle model, the grand partition function 3 and its subordinate functions become simplified, as is well-known [47], The zeroth-order Hamiltonian of the many-body perturbation theory using the MpUer-Plesset partitioning is one such case ... 

In this chapter, we consider a form of perturbation theory which is associated with the names of Rayleigh and hrddinger (RSPT). Since we are interested in obtaining a perturbation expansion for the correlation energy, we choose the Hartree-Fock Hamiltonian as our zeroth-order Hamiltonian. RSPT with this choice of was applied to N-electron systems in the early days of quantum mechanics by C. Moller and M. S. Plesset and hence is sometimes called Moller-Plesset perturbation theory (MPPT). 

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