Finite field perturbation theory

These properties have in the present work been computed using finite field perturbation theory (FFPT). Five field strengths were used — 0.01, — 0.005, 0.0,0.005, and 0.01 a.u. The energies were fitted to a polynomial of degree 4. Two sets of such calculations were performed one that includes all the 44 states (Case 1) and one that includes the ground state only (Case 2). The latter study will of course give the most accurate value for these properties. It should, however, be possible to use the Case 1 study to estimate the contribution of SOC to them. All calculations were performed at the equilibrium geometry (1.926 A). 

FINITE-FIELD PERTURBATION THEORY OF FERMI-CONTACT INTERACTIONS... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

From our experimental results and different models used in theoretical calculations using either CND0/2 (23-25, 37>38) and PCIL0 methods (26,27), or the electric field effect by IND0 finite perturbation theory (28), the following models can be supposed ... 

Very accurate values of the dipole and quadrupole polarizability for the equilibrium internuclear distance of HF can be found in a review article by Maroulis [71], calculated with finite-field Mpller-Plesset perturbation theory at various orders and coupled cluster theory using a carefully selected basis set. 

Fibrillin, calcium binding, 46 473, 474, 477 Fibulin-I, calcium binding, 46 473 Field desorption mass spectroscopy, 28 6, 21 Field effects, of astatophenols, 31 66 Fine structure, 13 193-204 Fingerprinting of polymetalates, 19 246-248 Finite perturbation theory, 22 211, 212 First transition series, substitution, transferrins, 41 423 26... 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

Once the reliability of CCSD(T) had been established, we could proceed with confidence to use it to predict vibrational frequencies for Be3 and Be. In order to obtain the best possible prediction to aid experimentalists, a full quartic force field was generated for each molecule [76], using finite differences of computed energies, and fundamental frequencies were obtained via second-order perturbation theory. In Table 5.7 we list the CCSD(T) fundamental frequndes and, for comparison, the CCSD, CCSD(T) and MRCI harmonic frequencies. 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

Uncoupled methods [sometimes called the sum over states (SOS) methods] do not include the field in the Hamiltonian but use a time-dependent perturbation theory approach.38-56 A sum over all excited states is used that requires values for dipole moments in ground and excited states and excitation energies to be evaluated. One must choose the number of states at which to terminate the series. It has been shown in several studies of second-order nonlinearities38 that the /8 values converge after a finite number of states are chosen. Furthermore, this approach intrinsically accounts for frequency dependence. 

During the past decade, theoretical calculations of hyperpolarizabilities have been performed to help synthetic chemists design optimum NLO structures. Although extremely accurate calculations are still out of reach, it is now possible to predict the influence of structural changes on the NLO coefficients. In the case of photochromes, theoretical calculations may be useful for predicting 3 values of thermally unstable colored forms. The theoretical methods generally employed to calculate molecular hyperpolarizabilities are of two types those in which the electric field is explicitly included in the Haniiltonian, frequently labeled as Finite Field (FF) and those which use standard time dependent perturbation theory, labeled Sum Over State (SOS) method. 

In another study of the polarizability and hyperpolarizability of the Si atom Maroulis and Pouchan6 used the finite field method with correlation effects estimated through Moeller-Plesset perturbation theory. Correlation effects are found to be small. 

Vibrational contributions to the a and (1 response functions of NaF and NaCl have been calculated by Andrade et al 5 at HF, MP and CC levels. The results obtained from perturbation theory are in agreement with those from the finite field method and demonstrate that the inclusion of vibrational effects is essential to get reliable electric response functions in these molecules. 

Besides the ultimate experimental determination of the properties, it is important to point out that the development of computational quantum procedures has also significantly boosted the search for NLO chromophores. There are basically two different computational approaches towards p (i) the sum over states (SOS) perturbation theory, and (ii) the derivative method, using the finite field (FF) procedure. 

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