Variational-perturbation calculations

In the present variation-perturbation calculations the first order corrections were expanded in 600-term ECG basis defined in equations (15) and (16). The components of the polarizability were computed from equation (11) using the optimized The optimization was performed separately for each component and intemuclear distance. The values of aj, (co) are arithmetic sums of the plus and minus components (equation (12)) computed from two separate first-order corrections. For a given component v (either or ), and are expanded in the same basis but, because they are solutions to two different equations (equation (9)) they differ in the linear expansion coefficients. The computed components of the static polarizability an(/ ) and a R) are drawn in Fig. 2 and their numerical values at selected intemuclear distances are listed in Table 1. 

The tensor elements of x can be determined from measurements of macroscopic magnetic susceptibility or evaluated from molecular orbital methods and approximate variation perturbation calculations. Recently, calculations of the magnetic quadrupole polarizability of closed-shell atoms, and magneto-electric susceptibilities of atoms, have been made. These matters, which relate to the behaviour of microsystems under the simultaneous action of an electric and a magnetic field, will be dealt with in detml in subsequent sections. 

Q. Rutkowski. Relativistic perturbation theory II. Qne-electron variational perturbation calculations. J. Phys. B At. Mol. Opt. Phys., 19 (1986) 3431 441. 

A reference calculation of the collision-induced dipole polarizability of Hc2 was reported by Cencek et al. ° These authors reported calculations of the interaction-induced properties via variation-perturbation calculations... 

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

The variational calculations were performed using the Alchemy II package [67] while the further perturbation calculations used a code derived from the original CIPSI module. Proper interfaces between the two programs were developed. 

In order to overcome the optimization process of the (hyper) polarizabilities calculations, we have been led to deeply study the perturbational and variational methods and in particular the variation-perturbation treatment introduced by Hylleras (20) since 1930. We will not develop here the theoretical framework of the recent study of N. El Bakali Kassimi (21). We propose criteria for generating adequate sets of polarization functions necessary to calculate (hyper) polarizabilities. 

We have first been concerned with the computational point of view. Through the calculation of the dynamic polarizability of CO, we have developed a method based on the conventional SCF-Cl method, using the variational- perturbation techniques the first-order wavefunction includes two parts (i) the traditional one, developed over the excited states and (ii) additional terms obtained by multiplying the zeroth—order function by a polynomial of first-order in the electronic coordinates. This dipolar... 

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

To evaluate ot theoretically, one must use perturbation theory. Ramsey did so (see Murrell and Harget, pp. 121-123), and found that o is the sum of two terms, a positive term od (called the diamagnetic contribution, since it decreases the applied field), and a negative term op (the paramagnetic contribution). The term op involves the usual perturbation-theory sum over excited states, and therefore is difficult to calculate however, one can use a variation-perturbation approach (Section 1.10) in op calculations. For molecular protons, od exceeds op, and aH is positive. 

Because of the difficulty in evaluating the paramagnetic contribution, ab initio calculations of NMR shielding constants are few in number and limited to rather small molecules. Jaszunski and Sadlej used a variational perturbation approach with an ab initio SCF wave function of H20 and found aH = 28.3 ppm. [M. Jaszunski and A. J. Sadlej, Theor. Chim. Acta, 27, 135 (1972).] The known proton screening constant in H2 is 26.6 ppm,6 and gaseous H20 shows a proton chemical shift of 3.6 ppm relative to gaseous H2 hence the experimental H20 proton shielding constant is 30.2 ppm. 

The influence of fluorine substitution on the torsional potential of 1,2-dithioglyoxal has been examined by Toro-Labbd140. However, this study has been carried out at a minimal basis set level, which yields quite poor molecular geometries. This implies that the trends shown for the torsional potentials are only of a qualitative value. The work of Cimiraglia and coworkers141 on variation-perturbation CIPSI calculations of the first excited states of 1,2-dithiete and 1,2-dithiin permits one to rationalize the main features of their electronic absorption spectra. 

C. A. Reynolds, J. W. Essex, and W. G. Richards, Chem. Phys. Lett., 199,257 (1992). Errors in Free-Energy Perturbation Calculations Due to Neglecting the Conformational Variation of Atomic Charges. 

Various semi-empirical methods have been compared for all properties in an important review by Klopman and O Leary, whilst Adams et < /. have compared the finite field, the variation, and the second-order infinite sum methods for the calculation of a in DNA bases. They find that the variation-perturbation method gives the most reliable results, but as their calculations were at the iterative extended Hiickel level there is no guarantee as to the generality of their conclusions. 

Initial results obtained for TPA and for photoelectron spectra of small systems, show that anharmonicity must be included in the calculation of EC factors to reproduce experiment [54, 77, 104]. However, it is difficult to treat larger anharmonic systems by means of perturbation theory. Such systems can be handled by applying the variation/perturbation methods of electronic structure theory that have been, and continue to be, extended to the vibrational Schrodinger equation as discussed earlier. The EC integrals tliat appear in the equations for resonant (hyper)polarizabilities may be calculated employing approaches like VSCF, VMP2, VCI and VCC, That will allow us to Include anharmonic contributions to all orders and thereby remove the intrinsic limitations of the perturbation expansion in terms of normal coordinates. 

The general discussion of Section 30, which is essentially a perturbation calculation, is not capable of very high accuracy, especially since it is not ordinarily practicable to utilize any central field except the coulombic one leading to hydrogenlike orbital functions. In this section we shall consider the application of the variation method (Sec. 26) to low-lying states of simple atoms such as lithium and beryllium. This type of treatment is much more limited than that of the previous section, but for the few states of simple atoms to which it has been applied it is more accurate. 

From the definition of molecular properties (29) and the above discussion it is clear that the proper theoretical framework for the description of molecular properties is variational perturbation theory. An excellent presentation of this approach is provided by Helgaker and Jprgensen [9]. Although they focus on the calculation of geometrical derivatives, the methods and proofs presented are straightforwardly extended to quasienergy derivatives, as shown by Christiansen eta/. [13]. 

The Breit-Pauli (BP) approximation [140] is obtained truncating the Taylor expansion of the Foldy-Wouthuysen (FW) transformed Dirac Hamiltonian [141] up to the (p/mc) term. The BP equation has the well-known mass-velocity, Darwin, and spin-orbit operators. Although the BP equation gives reasonable results in the first-order perturbation calculation, it cannot be used in the variational treatment. 

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