Vacuum Fermi

Here in eq. (38) "EpqfpQN a.pag is new Hartree-Fock operator for a new fermions (25), (26), operator Y,pQRsy>pQR a Oq 0s%] is a new fermion correlation operator and Escf is a new fermion Hartree-Fock energy. Our new basis set is obtained by diagonalizing the operator / from eq. (36). The new Fermi vacuum is renormalized Fermi vacuum and new fermions are renormalized electrons. The diagonalization of/ operator (36) leads Jo coupled perturbed Hartree-Fock (CPHF) equations [ 18-20]. Similarly operators br bt) corresponds to renormalized phonons. Using the quasiparticle canonical transformations (25-28) and the Wick theorem the V-E Hamiltonian takes the form... 

Further we want to study the nonadiabatic corrections to the ground state. Therefore /o> will be the unperturbed ground state wave function (we shall use Hartree-Fock ground state Slater determinant -Fermi vacuum) and % ) will be boson ground state-boson vacuum 0). 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

As concerns cluster amplitudes, if we employ the exact Hamiltonian in the normal-ordered-product form (31) with the /i-th configuration as a Fermi vacuum, the basic equation for the single-root wave operator (25) takes the form... 

Let us adopt I 0 > as a new vacuum state. We shall call it the Fermi vacuum state. 

With respect to the Fermi vacuum state I 0 > we can now define new creation and annihilation operators which in contrast to the X+, X operators shall be designated... 

From this definition it is evident that application of Yj to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in 14>0 >. The effect of YA on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in I 0>. The effect of YA" on the Fermi vacuum is the creation of a particle in the virtual spin-orbitals and finally, the effect of YA" is the annihilation of a particle in virtual spin-orbitals. Thus e.g., a singly excited Slater determinant I ) can be described as... 

Note that the first and second terms on the right-hand side of this equation are simply the spin-orbital Fock operator (in normal-ordered form), and the last two terms are the Hartree-Fock energy (i.e., the Fermi vacuum expectation value of the Hamiltonian). Thus, we may write... 

Ppor an explanation of -creation and -annihilation operators, see the earlier discussion of the particle-hole formalism in the section on The Fermi Vacuum and Particle-Hole Formalism. 

Questions concerning possible modifications to the descriptions of excited determinants and cluster operators Tm due to the renormalization and/or the use of normal product operators must be addressed. Excited determinants with respect to the Fermi vacuum can be written straightforwardly using creation and annihilation operators a singly excited determinant is given by... 

Application of the time-independent Wick theorem to the single-excitation operator X% Xiy present in both the description of singly excited determinants with respect to the Fermi vacuum and in the cluster operator T, gives... 

In describing many-body systems in their ground or low lying excited states it is convenient to redefine the vacuum state to contain the single particle states occupied in the ground state, . This is usually termed the Fermi vacuum. A set of creation and annihilation operators can be defined with respect to the Fermi vacuum as follows... 

Fermi-Vacuum Invariance in Multiconfiguration Perturbation Theory... 

Abstract We investigate the dependence of multiconfigurational perturbation theory framework on the choice of the Fermi-vacuum. A new formulation, based on a posteriori averaging is suggested. The averaged theory is invariant with respect to Fermi-vacuum choice but enhances the intruder effect. The performance of the averaged formulation is illustrated on the ethylene rotational potential curve. 

Keywords Multireference Perturbation theory Fermi-vacuum dependence Ethylene torsional barrier... 

In this study we consider the same problem in the framework of MCPT and propose a modification which restores the invariance to the choice of Fermi-vacuum. This involves calculating the perturbed quantities by all possible choices and constructing a weighted average. The number of parameters in the theory agrees with that of a Jeziorski-Monkhorst-type MRCC parametrization. The redundancy of a Jeziorski-Monkhorst parametrization however does not show up in the present approach due to the fact that perturbational amplitudes corresponding to different... 

In this account we first briefly summarize MCPT followed by an analysis on the extent of Fermi-vacuum non-invariance in different versions of the theory. We continue by presenting the approach to ensure Fermi-vacuum invariance and close with a numerical illustration. 

Using Eq. (9) in the expression for, orbital energies are independent from the Fermi-vacuum choice. It may be however appealing to substitute the Hartree-Fock density matrix... 

In contrast to the p-MCPT version, second-order u-MCPT energy shows explicit dependence on the Fermi-vacuum in the numerator. As illustrated in Section 3, dependence of second-order u-MCPT on the choice of HF) is more expressed in numerical terms than of p-MCPT. 

A simple way to remove dependence of the PT expressions on the Fermi-vacuum is to deliberately make every possible ehoice and form an average of the quantities obtained. Theoretical formulation may start by a linearized Jeziorski-Monkhorst-type parametrization of the wavefunction... 

Note, that L depends on K, and El — Eo may also be K dependent (e.g-in DK partitioning). For this reason the outer sum on K cannot be evaluated irrespective of L. Equation (15) is one of our working formulae, which ensures Fermi-vacuum independence of the energy, within the p-MCPT framework. 

It is worthwhile to examine a Fermi-vacuum invariant formulation with the use of basis vectors of u-MCPT also, e.g-to obtain a size-consistent second-order energy. This variant of the theory does not alter form (12) of the exact wavefunction, it only affects the expression of the excitation operator ... 

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