Atomic polarizability tensor

Electronic polarizability is often included in force fields via the use of induced dipoles. Assuming that hyperpolarization effects are absent, the induced dipoles respond linearly relative to the electric field. In this case, the induced dipole p on an atom is the product of the total electric field E and the atomic polarizability tensor a. 

At the same time, the formally independent particle nature of DFT allows the application of standard interpretative tools developed for the HF approach. This is true not only for the standard MuUiken population analysis, but also for more sophisticated schemes, like the Natural Bond Orbital (NBO) analysis [9], the Atomic Polarizable Tensor population [10], or the Atom in Molecule (AIM) approach [11]. These tools allow the use of familiar and well known models to analyze the molecular wave function and to rationalize it in terms of classical chemical concepts. In short, DFT is providing very effective quantum... 

In the general case, polarizability is anisotropic it depends on the position of the molecnle with respect to the orientation of the external electric field. To consider the orientation, a is expressed by a fnnction — atom polarizability tensor — that defines the indnced dipole moment for each possible direction of the electric field. This atom polarizability tensor a describes the distortion in the nuclear arrangement in a molecnle (i.e., the tendency of polarization in three dimensions). The polarizability determined in an experiment is an average polarizability it is the sum of polarizabilities in three principal directions that are collinear with the external field [59],... 

In the frame of characterizing the metal-insulator transition, the polarizability of linear ehains of equally spaced lithium atoms (Li , = 2, 4, 6, and 8) has been computed ab initio in the full configuration interaction limit.The perpendicular (to the chain direction) components of the per atom polarizability tensor, whieh depends little on the number of atoms, increases with the interatomic distance and tends monotonically towards the isolated atom polarizability value. On the other hand, the parallel component of the per atom polarizability displays a much different behavior (i) for short distanees, it inereases similarly to its perpendicular counterpart, (ii) then, it increases very quickly and this is magnified with larger number of Li atoms, and (iii) finally, for even larger distances, it decreases and tends to the isolated atom response. 

V. Relationship Between Atomic Polarizability Tensors and Valence Optical... 

VI. Effective Induced Bond Charges From Atomic Polarizability Tensors.261... 

AXa, Aya and Aza are the Cartesian displacement coordinates of atom a in a space fixed Cartesian coordinate system, and i, j and k are the respective unit vectors. Atomic polarizability tensors ax are third-rank tensor quantities which can be written 3 9 rectangular arrays ... 

Arranged in a row all ax tensors give the atomic polarizability tensor ax of the molecule... 

The elements of the atomic polarizability tensor can be evaluated by applying an ejqnession analogous to that used for dipole moment derivatives [Eq. (4.12)] [299]... 

In these equations oq and as are the matrices containing molecular polarizability derivatives with respect to normal and symmetry vibrational coordinates, respectively [Eqs. (8.41) and (9.3)], and Op is an array comprising polarizability derivatives with respect to molecular translations and rotations. The matrix product ag Bg represents die so-called vibrational atomic polarizability tensor Vq accounting for the changes in molecular polarizability with molecular vibrations. The Vq tensor for the entire molecule can be expressed as a juxtaposhion of individual atomic tensors ... 

The second term in Eqs. (9.75) and (9.76), die rotational atomic polarizability tensor reflects the contribution of molecular translation and rigid-body rotation to ax- The inclusion of the six external molecular coordinates in those equations - the diree translations Xy and X2, and the three rotations p, Py and P2, completes die set of molecular coordinates up to 3N. In diis vray polarizability dmivatives are transformed into quantities corresponding to a space-fixed Cartesian coordinate system. As already pointed out in section 4.1, the great advantage of such a step is that the imensity parameters defined in terms of a space-fixed coordinate system are independent on isotopic substitutions provided the symmetry of the molecule is preserved. This will be illustrated with an example in the succeeding section. By analogy with Eq. (9.77), die rotational polarizability tensor can be represented as... 

The relation is particularly usefiil in evaluatmg polarizability dmivatives wifii respect to symmetry coordinates from atomic polarizability tensors obtained through ab initio MO calculations. As mentioned, die APZT qipear as a standard outyut from programs for initio quantum mechanical calculations employing analytical derivative mediods. 

The elements of atomic polarizability tensors are interrelated by the following equadons... 

In this section an example of calculations employing the APZT formalism is presented. Atomic polarizability tensors for formaldehyde-do and foimaldehyde-d2 are evaluated. The initial data are taken from RHF/6-31G(d,p) db initio MO calculations [332]. Molecular geometry, static polarizability tensor and definition of symmetry coordinates are given in Table 9.11. The orientation of the molecule in the Cartesian space, definition of internal coordinates and numbering of atoms are shown in Fig. 3.1. 

The atomic polarizability tensors for H2CO and D2CO are obtained from Eq. (9.76). The elements of Bg and P matrices need to be rearranged to match the dimensions of a and Op, respectively. The ag Bg, Op p and ax tensors obtained for the two species are given in Tables 9.12 and 9.13. It can be seen that the atomic polarizability tensors ax for formaldehyde-do formaldehyde-d2 are, as expected, identical. 

RELATIONSHIP BETWEEN ATOMIC POLARIZABILITY TENSORS AND VALENCE OPTICAL FORMULATIONS OF RAMAN INTENSITIES... 

The atomic polarizability tensor ax for SO2 evaluated by employing Eq. (9.76) is shown as follows together with the vibrational and rotational polarizability tensors (in units A ) ... 

It can be seen that the atomic polarizability tensors obtained following the two alternative approaches as given in expressions (9.93) and (9.96) are equivalent. The equivalency is determined by Eq. (9.91). 

VI. EFFECTIVE INDUCED BOND CHARGES FROM ATOMIC POLARIZABILITY TENSORS... 

In diis section a method for interpretation of Raman intensities based on further transformations of atomic polarizability tensors is presented. The formulation was recently proposed by Ehidev and Galabov [333], A new molecular quantity - effective induced bond charge, Ok introduced. The effective induced bond charges are obtained from rotation-free atomic polarizability tensors following the strate as outlined by Galabov, Dudev and nieva [146] in the infrared case (Section 4.IV). The Ok parameters are expected to be associated with polarizability properties of valence bonds. 

The atomic polarizability tensor of an atom a is defined by expression (9.73). Arranged in a row all ax tensors form the polarizability tensor of the molecule, ax [E<1-(9.74)]. Its elements can be obtained via relations (9.75) and (9.76). As was discussed in Section 9.IV, atomic polarizability tensors are sum of two arrays (i) vibrational and (ii) rotational polarizability tensors [Eq. (9.79)]. The elements of atomic polarizability tensors are interconnected by the dependency conditions (9.84) and (9.85). The presence of such relations hampers die physical interpretation of these quantities. 

In vibrational analysis it is of particular importance to operate widi indepoident quantities that are associated with vibrational motions of die molecule only. Atomic polarizability tensors [Eq. (9.76)] can be corrected for non-vibradonal contributions following the procedure outlined below. The first step is to eliminate the contribution from the rotational atomic polarizability tensor [Eq. (9.78)]. After subtractmg Rq from both sides of Eq. (9.76) the vibrational atomic polarizability tensor is obtained... 

In this equation ax(v) is the atomic polarizability tensor free from any rotational contribution. Its elements are, however, still interrelated through the dependency condition (9.84). The problem can be solved if a set of bond displacement coordinates [Eqs. (4.96) and (4.97)] instead of atomic displacement coordinates is used. A rotation-free bond polarizability tensor is defined as... 

A representative series of molecules is selected to determine the trends of changes of the effective induced bond charges as defined by Eq. (9.102). The formulation developed has been applied in interpreting atomic polarizability tensors evaluated by HF/6-311+G(d,p) ab initio MO calculations [333]. A series of 17 molecules containing various bonds in different environment have been studied. The molecules are grouped as follows ... 

The symmetry coordinates employed have their usual form. Cartesian reference system and numbering of atoms and bonds are shown in Fig. 3.6. The atomic polarizability tensor ox obtained as a standard output of the ab initio calculations is as follows (in units of 10-30 C.mA0 ... 

As already mentioned, ax comprises contributions from two rotational atomic polarizability tensors UpP and PsB which have to be subtracted. These are given below (in units of 10 30 C.mAO ... 

This generalizes the expression for the energy of - E acquired by an atom of polarizability a in an external electric field E. For the Fourier component eE(io) exp(-io)t) of the incident radiation field, the frequency-dependent atomic polarizability tensor has the form... 

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