Analytic dipole moment derivatives

Consistent analytic evaluation of first, second and diird derivatives of the energy widi respect to nuclear coordinates has been an essential development in ab initio molecular orbital calculations of molecular properties. The theoretical basis of diese deveitqmients was created relatively early [166] and implemented first by Pople et d. [181]. Fence constants are evaluated in a much faster and accurate manner compared with die earlier methods based wholly or partially on finite-difference techniques. The mediodology is now developed for most types of molecular wave fimetions. 

The analytic determination of dipole moment derivatives by ab initio methods requires derivation of a complete formulation of coupled-perturbed Hartree-Fock equations by nuclear coordinate and electric field variations. 

The Hamiltonian operator perturbed by nuclear coordinate q and electric field f may be presented as follows [182] 

10) Hq is the unperturbed Hamiltonian while Xq and Xf are the respective perturbations. The perturbed integrals for basis set abnnic orbitals can diai be represented in a similar way. The respective formulas for one-electron integrals h v, overly integrals S v aod two-electron integrals (pv/pa) [ i(l)v(l)p(2)ot(2)] are 

For a standard atomic orbital basis set the perturbation Hamiltonian H q affects bodi one-and two-electron integrals while H f influences one-electron integrals only. 

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As expected, the effect of electron correlation on computed intensities has been initially treated by evaluating dipole moment derivatives through numerical differentiation [182,183]. The finite difference method of Komomicki and Mclver [178] offo another possibility. Simandiras et al. [184] have later developed a complete formulation for deriving analytic dipole moment derivatives at the second order Moller-Plesset (MP2) level. An alternative approach based on the configuration interaction (Cl) gradient concept [185] has been put forward by Lengsfield et al. [186]. 

As already underlined, no attempt to present a comprehensive account of ab initio calculations of infinred intensities will be made. A number of reviews on the subject are available [168,174-176,187,188], Besides, the progress in theoretical approaches and computational facilities makes earlier results of lesser current interest no matter how important they were at the time. The discussion will be concentrated on two main themes associated with these calculatimis (1) die influence of basis set and (2) die effect of electron correlation on computed intensities. We shall also limit our crmsiderations to results coming mosdy from analytic dipole moment derivative calculatirms. 

It is useful to define first die different basis sets of wave functions used in calculating analytic dipole moment derivatives and infrared intensities diat will be discussed in the present section. These are described in Table 7.1. 

With the development of analytical energy derivative methods135 l67, the calculation of vibrational frequencies (second derivatives of the energy with regard to atomic coordinates) and infrared absorption intensities (derivatives of the energy with regard to components of electronic field and atomic coordinates, i.e. dipole moment derivatives) both at the HF and correlation corrected levels has become routine168. There are six (two a " + four e)... 

The theory for the analytic evaluation of dipole moment derivatives will be considered first for the simplest case, that of closed-shell SCF wavefunctions. The electronic contribution to the dipole moment is... 

As with the closed-shell case, this matrix should be constructed from the derivative integrals in the atomic-orbital basis. Indeed, it is possible to solve the entire set of equations in the AO basis if desired. From these equations, it can be seen that properties such as dipole moment derivatives can be obtained at the SCF level as easily for open-shell systems as is the case for closed-shell systems. Analytic second derivatives are also quite straightforward for all types of SCF wavefunction, and consequently force constants, vibrational frequencies and normal coordinates can be obtained as well. It is also possible to use the full formulae for the second derivative of the energy to construct alternative expressions for the dipole derivative. 

The conclusion of this section is that analytic calculation of dipole moment derivatives is practicable and efficient for the most widely used ab initio methods, i.e. SCF, MC-SCF, Cl and Moller-Plesset perturbation theory. The author is not aware of any calculations of this type to date using correlated wavefunction methods, but it cannot be long before examples of these appear. 

The many body perturbation theory has been applied to obtain higher accuracy in ab initio calculations of molecular properties. Pople etal. [181] have developed analytic derivative methods at second order perturbation theory level (MBPT(2)). Simandiras et al. [184] have derived specific expressions for analytic determination of dipole moment derivatives at MBPT(2). Dierksen and Sadlej [229] have shown by applying finite field MBPT in studying dipole and quadruple polarizabilities of the CO molecule that fourth and even higher level of MBPT is required to achieve satisfactory results. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

Cl density method, which uses analytic derivatives of the wavefunction to compute the dipole moments, resulting in much more accurate predictions, as is illustrated in this case. You can request the Cl density by including either DensityaCI or DensityaCurrenI in the route section of a Cl-Singles calculation, n... 

The second derivatives can be calculated numerically from the gradients of the energy or analytically, depending upon the methods being used and the availability of analytical formulae for the second derivative matrix elements. The energy may be calculated using quantum mechanics or molecular mechanics. Infrared intensities, Ik, can be determined for each normal mode from the square of the derivative of the dipole moment, fi, with respect to that normal mode. 

To answer this question, let us first consider a neutral molecule that is usually said to be polar if it possesses a dipole moment (the term dipolar would be more appropriate)1 . In solution, the solute-solvent interactions result not only from the permanent dipole moments of solute or solvent molecules, but also from their polarizabilities. Let us recall that the polarizability a of a spherical molecule is defined by means of the dipole m = E induced by an external electric field E in its own direction. Figure 7.1 shows the four major dielectric interactions (dipole-dipole, solute dipole-solvent polarizability, solute polarizability-solvent dipole, polarizability-polarizability). Analytical expressions of the corresponding energy terms can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres (Suppan, 1990). These four non-specific dielectric in-... 

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