Inverse operator, expansion

The angular expansion functions have the parity (—1 )Jp under the inversion operation the parity parameter p has the values p = 1. 

The representation of hyperfine interactions in the form of multipoles follows from expansion of the potentials of the electric and magnetic fields of a nucleus, conditioned by the distribution of nuclear charges and currents, in a series of the corresponding multipole momenta. It follows from the properties of the operators obtained with respect to the inversion operation that the nucleus can possess non-zero electric multipole momenta of the order k = 0,2,4,..., and magnetic ones with... 

Note that no approximation has been made so far. The Breit-Pauli (BP) approximation [49] is introduced by expanding the inverse operators in the Schrodinger-Pauli equation in powers of (V — E) jlc and ignoring the higher-order terms. Instead, the BP approximation can be obtained truncating the Taylor expansion of the FW transformed Dirac Hamiltonian up to the (ptcf term. The one-electron BP Hamiltonian for the Coulomb potential V = Zr /r is represented by... 

This is an equation for the expansion coefficients of the large component only, and corresponds to solving the equations by standard matrix partitioning techniques. For actual calculations we would gain nothing, because we also have to find the expansion coefficients for the small component, but for purposes of analysis this form turns out to be convenient. In order to display the dependence on c more clearly, we can expand the inverse operator in the equation above by using the matrix relation... 

In the development of the Pauli Hamiltonian in section 17.1, truncation of the power series expansion of the inverse operator after the first term yielded the nonrelativistic Hamiltonian. In (18.1), the zeroth-order term is the Hamiltonian first developed by Chang, Pelissier, and Durand (1986), often referred to as the CPD Hamiltonian. The name given by van Lenthe et al. is the zeroth-order regular approximation, ZORA, which we will adopt here. The zeroth-order Hamiltonian is... 

In the derivation of the ZORA equation, we made the assumption that < 2mc so that we could do the expansion of the inverse operator. However, the ZORA equation has a valid spectrum for all E. It is therefore not necessarily the case that the use of a truncated expansion is invalid outside the strict radius of convergence. We could... 

However, this is only a cosmetic change, because the superoperator resolvent is an inverse operator and is only defined through its series expansion in Eq. (3.148). The way in which we can proceed, is to find a matrix representation of the superoperator resolvent. In order to do that we need a complete set of basis vectors like in the normal Hilbert space. However, the vectors in the superoperator formalism are operators and we therefore need a complete set of operators. Such a set of operators hn consists of a complete set of excitation and de-excitation operators with respect to the reference state This means that all other states of the system or all excited states of... 

In the second section of this chapter, we shall employ the partitioning technique to develop various types of perturbation theory, including Rayleigh-Schrddinger perturbation theory and Brillouin-Wigner perturbation theory. This involves the expansion of the inverse operators which occur in the effective Hamiltonian operator and other operators obtained by the partitioning technique. Different expansions lead to different types of perturbation theory. 

We are now in a position to obtain perturbation expansions by expanding the inverse operator in the effective Hamiltonian, the wave operator and the reaction operator. We begin, as we did in our discussion of the partitioning technique, by considering the case of a single-reference function and then turn our attention to the multi-reference function case. 

Different forms of perturbation theory can be obtained by expanding the inverse operator in eq. (2.79), i.e. ( - QSiQY. This operator is assumed to exist. It can be written as an infinite expansion using the operator identity... 

The solution of the Schrddinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an altemative insight in the mathematical structure of perturbation theory, particularly with respect to the bracketing theorem and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately. 

Thus, a generic hermitian operator can be expressed in terms of the representation of the group Hn. The existence of this expansion is demonstrated in Appendix 1. There, we also prove that the functions B(g) have to be obtained as the inverse Fourier transform of the functions Bw (q,p) of the phase-space variables (q,p) = qi,. . ., qn,Pi, , Pn), which are associated to the quantum operators, B, via the Wigner transform [10]. Since we are considering hermitian operators of the form B X, IiD J, the coefficients in eq.(22) are... 

On the other hand, the inverse of the exponentiated excitation operator, e T is also an excitation operator, as can be seen from its power series expansion. 

In the second equality we have expanded the coordinate deviation <5x in normal modes coordinates, and expressed the latter using raising and lowering operators. The coefficients are defined accordingly and are assumed known. They contain the parameter a, the coefficients of the normal mode expansion and the transformation to raising/lowering operator representation. Note that the inverse square root of the volume Q of the overall system enters in the expansion of a local position coordinate in normal modes scales, hence the coefficients scale like... 

In the presence of a (Coulomb) potential no closed solution is possible. However an expansion in powers of is formally straightforward. As shown in the Appendix, the perturbative construction of the transformation to a diagonal operator requires the inversion of commutators with the unperturbed Hamiltonian Hq. In our case, where Do = (3mc the commutator inversion is particularly simple [12, 13]. The solution of the equation... 

In the asymptotic region, i.e. for large intermonomer separations R, the interaction energy is well described by the polarization terms alone. Moreover, at such distances one can make an additional approximation and represent the operator V in terms of its multipole expansion containing terms inversely proportional to powers of R. As R increases, the interaction energy will eventually be well represented by just the term with the lowest power of l//f. For polar dimers such as water, the lowest power equal to three is coming from the electrostatic interactions and the water dimer potential... 

---