Matrix, dipole moment integrals

In the program it is only necessary to modify the matrix elements of the one-electron part of tire Hamiltonian h j by adding the dipole moment integrals ... 

Figure 7,4 Top Replica HNC predictions for the stability limits of the homogeneous isotropic phase of Stockmayer fluids adsorbed to disordered DHS matrices of density p,n = 0.1. Curves are labeled according to the reduced matrix dipole moment fJ m/ sTocr (the pure HS matrix corresponds to = 0). Bottom Dielectric constant of a dense adsorbed fluid as a function of the matrix dipole moment T = 0.5, f) = 0.7, Pm = 0.1). The inset shows the integrated blocking part of the dipole dipole correlation function.

The electronic contribution to the dipole moment is thus determined from the density matrix and a series of one-electron integrals J dr< (-r)0. The dipole moment operator, r, h.)-components in the x, y and z directions, and so these one-electron integrals are divided into their appropriate components for example, the x component of the electronic contribution to the dipole moment would be determined using ... 

Similar considerations apply to the electric dipole moment the derivatives of the dipole integrals can be easily obtained whilst the derivatives of the density matrix require the use of coupled Hartree-Fock theory (e.g. Gerratt and Mills, 1968). 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

According to the results of the last section, if the integrals in (3.48) all vanish, then the probability for a transition between states m and n is zero. Actually, (3.47) is the result of several approximations, and even if the electric dipole-moment matrix elements vanish, there still might be some probability for the transition to occur. 

The pattern of intensities in Fig. 4.9 deserves mention. The intensities of absorption lines are proportional to the population of the lower level, and to the square of the dipole-moment matrix element (4.97). It turns out that for vibration-rotation transitions in the same band, the integral (4.97)... 

The electronic contribution to the dipole moment is thus determined from the density matrix and a series of one-electron integrals r) . The dipole moment operator, r, has... 

This is all we can say in principle about calculation of the dipole moment in the Hartiee-Fock approximation. The rest belongs to the technical side. We choose a coordinate system and calculate all the integrals of type (xfckx/) ie-, ixk xxi) (Xk yxi) (Xk zxi)- The bond order matrix P is just a by-product of the Haitree-Fock procedure. 

According to Fermi s golden rule [40, 42], the integral intensity A of the absorption band of the normal mode is proportional to the probability per unit time of a transition between an initial state i and a final state j. Within the framework of the first (dipole) approximation of time-dependent perturbation quantum theory [46, 65], this probability is proportional to the square of the matrix element of the Hamiltonian H = —E p, where E is the electric field vector and p is the electric dipole moment, resulting in the absorption... 

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. 

SO that, in addition to the density matrix, we need only evaluate the set of one-electron integrals fi h v) to calculate one-electron expectation values. We will use the dipole moment to illustrate such a calculation. 

---