Polarization propagator derivative

The above comprises the derivation of the expression for the PES of the complex system which is not only free from the necessity to recalculate the wave function of the classical subsystem in each point, but formally not requiring any wave function of the M-system at all, since the result is expressed in terms of the generalized observables - one-electron Green s functions and the polarization propagator of the free M-system. Reality is of course more harsh as the necessary quantities must be known for a system we know too little about, except the initial assumption that its orbitals do exist. Section 3.5 will be devoted to reducing this uncertainty. 

In this chapter, we will not be concerned with the detailed expressions of the response functions that we find for the standard electronic structure methods in theoretical chemistry. However, we will briefly outline the basic elements in two alternative formulations of response tlieory, namely the polarization propagator and the quasi-energy derivative approaches. 

The expression for the cubic response function is given in Eq. (2.60) of Olsen and Jorgensen (1985). All the propagators that are derived from response theory are retarded polarization propagators. The poles are placed in the lower complex plane. This is specified through the energy variables Ei+itj and 2 + ii . The Pjj operator in Eq. (35) permutes Ei and 2 and it is assumed that the - 0 limit must be taken of the response functions. 

Eq. (58) represents the starting point for all approximate propagator methods. Even though in the derivation we only discussed the linear response functions or polarization propagators, a similar equation holds for the electron propagator. The equation for this propagator has the same form but there are differences in the choice of h and in the definition of the binary product (Eq. (52)), which for non-number-conserving, fermion-like operators should be... 

From Eqs (67) and (114) it follows that the MCSCF polarization propagator is (for a detailed derivation see J irgensen and Simons, 1981, Chapter 6.E.3))... 

Also the algebraic diagrammatic construction (ADC) method that was discussed in Section VI has been applied to the polarization propagator (Schirmer, 1982). Diagrammatic rules, rather than the analytic derivation used in SOPPA, are applied to formulate the second-order ADC(2) and basically the same approximation is obtained. 

C-C Coupling constants calculated using second order polarization propagator approach (SOPPA) are in a good agreement with the available experimental data and can be used for conformational analysis in the N-containing carbonyl derivatives (Figure 12.48). ... 

J.E. Del Bene, 1. Alkorta, I.J. Elguero, Systematic comparison of second-order polarization propagator approximation (SOPPA) and equation-of-motion coupled cluster singes and doubles (EOM-CCSD) spin — spin couphng constants for molecules with C, N, and O double and triple bonds and selected F-substimted derivatives, J. Chem. Theor. Comput. 5 (2009) 208—216. 

Exercise 3.7 Fill in the missing steps in the derivation of Eq. (3.106), which proves that the polarization propagator depends only on the time difference t — t. ... 

Exercise 3.8 Derive the expression for the polarization propagator in the frequency domain, Eq. (3.110), from the expression in the time domain, Eq. (3.107). 

This equation, which is the perturbation theory expansion of an expectation value in the presence of a static field P in static response functions or polarization propagators, is another way of writing Eq. (3.36). Therefore, we can identify the static response function as the first derivative of the first-order correction to a perturbed expectation value, i.e. 

Before we continue in the derivation of a matrix representation of the polarization propagator, we want to mention that by taking the zero-frequency limit of the equation of motion in the frequency domain, we obtain the following relation between a polarization propagator and a ground-state expectation value... 

In Part III we will come back to these expressions and evaluate the derivatives of approximate wavefunctions. However, here we will use the response formalism as developed in Section 3.11. Using Eq. (3.116) we can express the derivatives of the perturbation dependent expectation value in terms of polarization propagators or linear response functions and thus obtain for the tensor components of the polarizabilities... 

The derivation of the induced contribution, on the other hand, is very similar to the derivation for the magnetizability. We could start from the definition of the rotational g tensor as first derivative of the rotational magnetic moment, Eq. (6.8), which would then be the induced contribution to it, and use the response theory formalism of Section 3.11. Using Eq. (3.116) we could express the derivatives of the induced rotational magnetic moment in terms of a polarization propagator and ground-state expectation value. Here we will, however, make use of the definition as second... 

We are going to rewrite the three linear response functions now as ground-state expectation values similar to the derivations in Section 5.9. However, here we wiU not proceed via the sum-over-states expressions for the response function, but want to illustrate an alternative approach via the equation of motion of the polarization propagator for zero frequencies, Eq. (3.141). Recalhng that O p is the canonical conjugate... 

Recalling that the frequency-dependent polarizability is related to the (( fia A/3 ))w propagator, Eq. (7.26), we can express the even dipole oscillator strength sums also as derivatives of this polarization propagator, i.e. 

Exercise 10.3 Derive the partitioned form of the matrix representation of the polarization propagator, Eq. (10.15), using the relation for the inverse of a blocked matrix, Eq. (10.14). 

This approximation is better known as the time-dependent Hartree—Fock approximation (TDHF) (McLachlan and Ball, 1964) (see Section 11.1) or random phase approximation (RPA) (Rowe, 1968) and can also be derived as the linear response of an SCF wavefunction, as described in Section 11.2. Furthermore, the structure of the equations is the same as in time-dependent density functional theory (TD-DFT), although they differ in the expressions for the elements of the Hessian matrix E22. The polarization propagator in the RPA is then given as... 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

Table 13.2 Comparison of different polarization propagator (Dalskov and Sauer, 1998) and analytical derivative methods (McDowell et al., 1995) for the calculation of static dipole polarizabihties a (in units of e aoE ) using the medium-size polarized basis sets (Sadlej, 1988, 1991a Andersson and Sadlej,...

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