Perturbation theory wavefunction expressions

The second-order correction to the overlap matrix S consists of a similar term with the second-order correction to the density matrix, Eqs. (9.116) and (9.117), as the renormalization contribution the A-matrix. This is to be expected as the overlap matrix is related to the norm of the second-order Mpller-Plesset perturbation theory wavefunction, which can be expressed in terms of the second-order correction to the density matrix as... 

The perturbed energies and wavefunctions for the i-th system state can be expressed in a similar way as in scalar perturbation theory ... 

In other words, the diagonal elements of the perturbing Hamiltonian provide the first-order correction to the energies of the spin manifold, and the nondiagonal elements give the second-order corrections. Perturbation theory also provides expressions for the calculation of the coefficients of the second-order corrected wavefunctions l / in terms of the original wavefunctions (p)... 

It has been well known for some time (e.g. [36]) that the next component in importance is that of connected triple excitations. By far the most cost-effective way of estimating them has been the quasiper-turbative approach known as CCSD(T) introduced by Raghavachari et al. [37], in which the fourth-order and fifth-order perturbation theory expressions for the most important terms are used with the converged CCSD amplitudes for the first-order wavefunction. This account for substantial fractions of the higher-order contributions a very recent detailed analysis by Cremer and He [38] suggests that 87, 80, and 72 %, respectively, of the sixth-, seventh-, and eighth-order terms appearing in the much more expensive CCSDT-la method are included implicitly in CCSD(T). 

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

The notation in Eqs. (6a) and (6b) corresponds to the usual notation of perturbation theory [18] in which it is understood that the derivatives of the Hamiltonian as well as all variational and nonvariational parameters (i.e., including the dependence of the wavefunction on the perturbations) are taken. Some authors prefer to write Eqs. (6a) and (6b) as total derivatives of E in order to indicate that all dependencies of E on the perturbation parameters must be considered [19,20], Ultimately one needs to consider an explicit energy expression in which no hidden dependencies on any of the perturbations are left. This also includes a possible dependence of the basis set on the perturbation. 

In Kohn-Sham DFT based approaches, expressions that are of similar structure as Eqs. (9a) and (9b) are obtained, but in the form of contributions from all occupied Kohn-Sham MOs The excited-state wavefunctions are at the same time formally replaced by the unoccupied MOs, and the many-electron perturbation operators /T(M41, etc. by their one-electron counterparts //(M-41, etc. Orbital energies e and ea formally substitute the total energies of the states (see later). Thus, similar interpretations of NMR parameters can be worked out in which the highest occupied MO-lowest unoccupied MO gap (HLG) plays a highly important role. It must be emphasized, though, that there is no one-to-one correspondence between the excited states of the SOS equations and the unoccupied orbitals which enter the DFT expressions, nor between excitation energies and orbital energy differences, i.e., there are no one-determinantal wavefunctions in Kohn-Sham DFT perturbation theory which approximate the reference and excited states. 

For an elaborated analysis of the relations between structure and hyperpolarizabilities, one has to start from the electronic wavefunctions of a molecule. By using time-dependent perturbation theory, sum-over-states expressions can be derived for the first and second-order hyperpolarizabilities j3 and y. For / , a two-level model that includes the ground and one excited state has proven to be sufficient. For y the situation is more complicated. 

The theoretical understanding of the interaction between molecules at distances where the overlap is negligible has been well established for some years. The application of perturbation theory is relatively straightforward, and the recent work in this area has consisted in the main of the application of well-known techniques. In the case of neutral molecules, the first non-zero terms appear in the second order of perturbation, so that some method of obtaining the first-order wavefunction, or of approximating the infinite sum in the traditional form of the second-order energy expression, is needed. 

There are two major ways to view tlie vibrational contribution to molecular linear and nonlinear optical properties, i.e. to (hyper)polarizabilities. One of these is from the time-dependent sum-over-states (SOS) perturbation theory (PT) perspective. In the usual SOS-PT expressions [15], based on the adiabatic approximation, the intermediate vibronic states K, k> are of two types. Either the electronic wavefunction... 

Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

Quantum-mechanical expressions for the polarizability and other higher-order molecular response tensors are obtained by taking expectation values of the operator equivalent of the electric dipole moment (2.5) using molecular wavefunctions perturbed by the light wave (2.4). This particular semi-classical approach avoids the complications of formal time-dependent perturbation theory it has a respectable pedigree, being found in Placzek s famous treatise on the Raman effect [9], and also in the books by Born and Huang [lO] and Davydov [ll]. Further details of the particular version outlined here can be found in my own book [12]. 

Again, this is just the leading term in the expansion. More complete formulae can be found in Toyama et al., Krohn et al. and Riley et aV Particular expressions for the vibrational and rotational effects in symmetric and spherical tops have been given by Fowler. These expressions are all based upon perturbation theory, and if extended to very high vibrational levels will eventually break down. Under these circumstances, it will be better to solve explicitly for the vibrational wavefunctions and to evaluate the expectation values of the properties by direct integration. 

The first-order wavefunction in Rayleigh-Schrddinger perturbation theory for a single perturbation is given by equation (24). Using equation (38), this may be written as the sum-over-states expression... 

The present paper is aimed at developing an efficient CHF procedure [6-11] for the entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up to the 5-th rank. Owing to the 2n+ theorem of perturbation theory [36], only 2-nd order perturbed wavefunctions and density matrices need to be calculated. Explicit expressions for the perturbed energy up to the 4-th order are given in Sec. IV. 

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