First-order electrical properties

In this Section, the derivation of useful expressions for the calculation of first-order properties at the quasi-relativistic level of theory will be outlined. The electric field gradient at the nucleus is chosen to represent first-order electrical properties. The relativistic corrections to the electric field gradients are large since the electric field gradient operator is proportional to r. The electric field gradient operator is thus mainly sampling the inner part of the electronic density distribution. 

First-order electrical properties can conveniently be determined from expressions derived by using gradient theory. The Hamiltonian is augmented with an operator representing the studied property. By calculating the first derivative of the total energy with respect to the strength parameter of the property operator, one obtains an expression for the calculation of the specified first-order property. 

In the previous Subsection, the derivation of expressions for the calculation of the electric field gradient at the quasi-relativistic level of theory has been outlined. Similar expressions must be used in order to obtain accurate values for other first-order electrical properties at quasi-relativistic level of theory. The expressions obtained in the present derivation show that at the quasi-relativistic level of theory, first-order properties must not be calculated as pure expectation values of the nonrelativistic property operator, but other operators also appear in the expressions. This is the so called picture-change effect previously discussed in several articles [71-76]. 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

Five large basis sets have been employed in the present study of benzene basis set 1, which has been taken from Sadlej s tables [37], is a ( ()s6pAdl6sAp) contracted to 5s >p2dl >s2p and contains 210 CGTOs. It has been previously adopted by us in a near Hartree-Fock calculation of electric dipole polarizability of benzene molecule [38]. According to our experience, Sadlej s basis sets [37] provide accurate estimates of first-, second-, and third-order electric properties of large molecules [39]. 

We shall later (section 4.8) study the relativistic corrections to properties. At this point we are interested in their nrl. We shall see that for electric properties one gets the correct nrl automatically from the Schrodinger equation. The nrl of magnetic properties is more subtle. The result for a first-order magnetic property is e.g. 

First and second order electrical property LiH molecule... 

An important application of theoretically calculated electric field gradients at the nucleus is in the determination of nuclear quadrupole moments. The interaction between the nuclear quadrupole moments and the electric field gradient at the nucleus can be measured with unprecedented accuracy (compared to other experimental approaches for determining nudear quadrupole moments) in microwave spectroscopy. These interactions give rise to the fine structure in the rotational spectrum of the molecule. Highly accurate estimates for the electric field gradient at the nucleus can often be calculated theoretically (since this is a first-order molecular property). By combining these theoretical results with experimental observations, accurate values of the nuclear quadrupole moments can be obtained, and this approach has been used to revise previously estimated nudear quadrupole moments (see, e.g., KeUo and Sadlej 1998). 

We have next to consider the measurement of the relaxation times. Each t is the reciprocal of an apparent first-order rate constant, so the problem is identical with problems considered in Chapters 2 and 3. If the system possesses a single relaxation time, a semilogarithmic first-order plot suffices to estimate t. The analytical response is often solution absorbance, or an electrical signal proportional to absorbance or to another physical property. As shown in Section 2.3 (Treatment of Instrument Response Data), the appropriate plotting function is In (A, - Aa=), where A, is the... 

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

The polarizability expresses the capacity of a system to be deformed under the action of electric field it is the first-order response. The hyperpolarizabilities govern the non linear processes which appear with the strong fields. These properties of materials perturb the propagation of the light crossing them thus some new phenomenons (like second harmonic and sum frequency generation) appear, which present a growing interest in instrumentation with the lasers development. The necessity of prediction of these observables requires our attention. 

Nonlinear second order optical properties such as second harmonic generation and the linear electrooptic effect arise from the first non-linear term in the constitutive relation for the polarization P(t) of a medium in an applied electric field E(t) = E cos ot. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

The average polarizability a, defined by equation 9, is a global property, which pertains to a molecule as a whole. It is a measure, to the first order, of the overall effect of an external electric field upon the charge distribution of the molecule. We are unaware of any experimentally determined a for the molecules that are included in this chapter. However, they can be estimated using equation 12 and the atomic hybrid polarizabilities, and corresponding group values, that were derived empirically by Miller. These were found to reproduce experimental molecular a with an average error of 2.8%. The relevant data, taken from his work, are in Table 7. 

The first-order property for which most experimental20-28 and theoretical results exist is the (electric) dipole moment. The literature on calculations of dipole moment is almost as large as that on wavefunctions. 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

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