Electric dipole correlation functions

According to the formalism we developed in Chapter 1, the probability amplitude for an El one-photon transition from state k to state m is 

In the limit of long times t co k, the co-dependent factor in Eq. 8.3 approaches a constant times the Dirac delta function, 

The transition probability is clearly proportional to the time duration t for which the external field is applied. It is therefore meaningful to define the transition probability per unit time. 

This expression coincides with the well-known Golden Rule formulation [1] of the molecular transition probability under the external perturbation W= - iEo. 

We now wish to generalize this expression to a system of molecules at thermal equilibrium. Let p and be the probabilities that a molecule will be found in state k and in state m, respectively. The rate of energy loss from the radiation 

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I (6) is then given by the Fourier transform of the electric dipole correlation function For a single molecule, the quantity /i(0) /i(t) gives the... 

One of the first applications of RQMC was to the rotational dynamics of carbonyl sulfide (OCS) molecules solvated in helium clusters, for cluster sizes (tV = 3,10) [42]. This and related work, described shortly, rest on the absorption spectrum given by the Fourier transform of the reptilian imaginary time electric dipole correlation function. Similarly, the optical activity is extracted from the autocorrelation of the molecular orientation vector. This work by Moroni and coworkers and/or Boroini and co-workers was closely followed by several other investigations of rotational dynamics in doped clusters, summarized as follows ... 

Without the external field, the Stockmayer fluid near the wall exhibits symmetric density oscillations that die out as they reach the middle of the film. Near the surface, the fluid dipoles are oriented parallel to the walls. Upon turning on the electric field, the density profile of the Stockmayer fluid exhibits pronounced oscillations throughout the film. The amplitude of these oscillations increases with increasing field strength until a saturation point is reached at which all the fluid dipoles are oriented parallel to the field (perpendicular to the walls). The density profile remains symmetric. The dipole-dipole correlation function and its transverse [] and longitudinal [] com-... 

An alternative approach to DS study is to examine the dynamic molecular properties of a substance directly in the time domain. In the linear response approximation, the fluctuations of polarization caused by thermal motion are the same as for the macroscopic rearrangements induced by the electric field [27,28], Thus, one can equate the relaxation function < )(t) and the macroscopic dipole correlation function (DCF) V(t) as follows ... 

On the assumption of Nee and Zwanzig, f falls into two parts, one of whidi they determine as a known function of e. The other is 2kT times the reciprocal of the transformed memory function for the dipole correlation function yg of the motion in the absence of the electric dipole interactions, so... 

To determine an accurate electric dipole moment function (EDMF) for the X n state of OH requires a very high level of correlation treatment, since it is necessary to properly account for the O" character in the wave function. To calibrate approximate methods, an FCI dipole moment was computed at five representative r values using a [4s3p2d/2slp] Gaussian basis set. Of the variety of approximate methods compared with the FCI, the CASSCF/MRCI treatment reproduced the F( I results best, with an error of only 0.001 Aq in the position of the dipole moment maximum. The MRCI spectroscopic constants are also in excellent agreement with the FCI. At this level of correlation treatment, it did not make a substantial difference... 

In order to calculate the dipole correlation function, let us consider an ensemble of N identical rigid molecules, each possessing a dipole moment m. To describe the orientation of a molecule in space, two coordinate systems are introduced. The laboratory frame of reference (XYZ), which we will call LF, is traditionally defined as having the Z-axis in the direction of the probing electric field. The molecular frame of reference fixed within the molecule (xyz), which we shall refer to as mF, usually has axes chosen along the principal axes of the moment of inertia tensor (or any other molecular tensor). The orientation of the molecule is then given by the orientation of mF with respect to LF, which is determined by a set of Eulerian angles Q = a, jS, (see Fig. 4.4). The molecular dynam-... 

The probing electric field senses the component of the molecular dipole moment in the direction of this field. One is, therefore, interested in calculating the dipole correlation function of the dipole moment resolved along the Z-axis of LF. It is convenient for this purpose to build the first-rank irreducible spherical tensor from the principal components of u in mF ... 

In this form, one says that the time dependence has been reduce to that of an equilibrium averaged (n.b., the Zj pi i I I i>) time correlation function involving the component of the dipole operator along the external electric field at t = 0 ( Eo p ) and this component at a different time t (Eq p (t)). 

Raman scattering depends on the time correlation function of the many-body polarizability of the liquid, collective dipole moment. In the case of Raman scattering, an external electric field (from a laser) generates an induced collective dipole in the liquid ... 

The polarizability time correlation function of a liquid will therefore consist of a single-molecule contribution, a DID contribution, and a cross term between the two. The cross term is negative for intermolecular modes, since on the whole the field generated by the molecular dipoles tends to oppose the applied electric field, thus reducing the many-body polarizability of the liquid (23). 

The functions X(Q) and Y(fl) are specified by the choice of the particular experiment. Prominent orientational correlation functions result when setting X(Q) = K(Q) = P/(cos0), where P/ is the Legendre polynomial of rank / and the angle 0 specifies the orientation of the molecule with respect to some fixed axis. For example, consider a molecule that possesses a vector property, say the molecular electric dipole p = pu, (u is a unit vector). Then, one defines the dipole autocorrelation function g (t) = (u,(t)u,(0)). Similarly, one defines a correlation function gilt) for second rank tensorial molecular properties. In general the normalized (g/(0) = 1) orientational correlation function of rank l is given by... 

An alternative way of calculating the static polarizability is to evaluate the induced electric dipole moment by applying an external electric field in the SCF calculations, as described in section III. Variation of the induced dipole moment as a function of the applied electric field in the SCF calculations for Ceo is shown in Fig. 21 for the Hedin-Lundquvist exchange-correlation potential and an extended basis set. The calculations were performed using an electric field of strength up to 0.032 a.u. corresponding to 0.16-10 Vm . ... 

The connection of dipole correlation with angular-momentum correlation cannot be quite so direct for a non-linear molecule. Rahman and Stillinger have published a detailed molecular-dynamics study of a water model containing 216 model molecules in a cube with periodic boundary conditions, at a density of 1 kg dm . The short-time behaviour of the molecules is dominated by librations at 1—2 x 10 Hz. They ve Cole-Cole plots of the electric permittivity estimated by applying an analysis due to Nee and Zwanzig to the computed molecular-correlation function. However, this application is uncertain in view of the... 

For electric dipole moments it is clear that the agreement between theory and experiment is not improved when the basis set is extended from minimal to split valence shell. It should be noted that calculated values at the minimal level, in general, are lower than experiment, while the 4-31G results are too large. A considerable improvement is obtained when d functions are added to the first-row atoms. Limited data suggest that the 6-31G values are close to those that would be obtained if perfect molecular orbitals were used. Sizable discrepancies, however, still remain between such Hartree-Fock values and experiment, indicating that electron correlation can make substantial contributions to electric dipole moments. 

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