Derivatives first anharmonicity

The MCSCF gradient expression was first given by Pulay (1977). The MCSCF Hessian and first anharmonicity expressions were derived by Pulay (1983) using a Fock-operator approach, and by Jprgensen and Simons (1983) and Simons and Jorgensen (1983) using a response function approach. 

To summarize, the first anharmonicity may be evaluated for an MCSCF wave function with second and third derivative integrals in the AO basis, first derivative integrals in the MO basis with two general and two active indices, and undifferentiated integrals with three general and one active indices. 

All contributions to the molecular derivatives involving the higher electronic derivatives [Eqs. (59) and (60)] may be treated as direct linear transformations and calculated in terms of inactive Fock matrices containing multiply one-index transformed integrals. For example, the first anharmonic-ity contains the term... 

Let us summarize. The calculation of Cl first anharmonicities requires no storage or transformation of second and third derivative two-electron integrals, but the full set of first derivative MO integrals is needed. One must construct and transform one set of effective density elements for third derivative integrals and 3M — 6 sets of effective densities for second derivative integrals. In addition to the 3N — 6 MCSCF orbital responses k(1) and the Handy-Schaefer vector Cm needed for the Hessian, the first anharmonicity requires the solution of 3JV — 6 response equations to obtain (1). 

A whole range of different properties can be obtained by choosing other interaction operators W t) and thus other V and A operators. Examples are given by Olsen and Jorgensen (1985) and include a diversity of properties such as derivatives of the dipole polarizability, the B term in magnetic circular dichroism, the first anharmonicity of a potential energy surface, two-photon absorption cross sections and derivatives of the dynamic hyperpolarizability. 

It is useful to understand the effect of the employed derivatives on the properties of interest [50]. We specifically consider, as a test case, the components Pxxz and Yxxzz. The anharmonicities will be described by (mnop) [50], where m denotes the mechanical and nop defines the electrical anharmonicity [51]. m indicates the use of harmonic (m = 0), cubic (m= 1), and quartic (m = 2) approximation to the potential. Dipole moment derivatives n = 0,1,2,3, that is no derivatives first derivatives first and second derivatives first, second and third derivates are taken into account, respectively. Polarizability derivatives o = 0,1,2 and first hyperpolarizability derivatives p = 0,l the notation is similar to that given for the dipole moment derivatives [52]. We will consider some specific examples. It... 

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

The repulsive frequency shift, Av0, is expressed explicitly in terms of the first and second derivatives of the excess chemical potential (equation 2) along with the vapor phase vibrational transition frequency, vvib, equilibrium bond length, re, and harmonic and anharmonic vibrational force constants, f and g (232528). 

Note that the value of the anharmonicity constant can thus be derived from a comparison between the fundamental n = 0 to n = 1 absorption band frequency and the first and subsequent overtone band positions. Given the form of the Morse Curve, Xe can be calculated from De, and hence an estimate of the bond dissociation energy... 

In the harmonic approximation only the term n = 1 is different from zero. It is then the anharmonicity that allows the coupling between the initial state and final states n > 1. The second-order derivative of the Hamiltonian does not pose any particular problem and can be evaluated in the same way as the first derivative (see above). 

Rotational constants obtained for both the ground and the three first excited vibrational states allowed one to derive the equilibrium molecular structures of GeF2 (re = 1.7321 A, 6>e = 97.1480211) and GeCl2 (re = 2.169452 A, <9e = 99.8825°285). From measurements of the Stark effect the dipole moment of GeF2 has been determined to be 2.61 Debye283. The harmonic and anharmonic force constants up to the third order have been obtained for both molecules and reported too283,285. 

The first derivative L tensor elements, are determined as the elements of the L matrix from the preliminary harmonic calculation. The second and third derivative L tensor elements have been determined in one of two ways in the literature of anharmonic calculations. The first method involves setting up an... 

The observed frequencies (ly) have a lower energy than the harmonic vibrations (Wf). As a first approximation, the following relation involving one anharmonicity constant Xe can be derived v = uj. (] 2 x ). Thus, force constants derived from experimentally... 

Quadratic terms in die property expansions are considered to be first-order in electrical anharmonicity, cubic terms are taken to be second-order, etc. Similarly, cubic terms in the vibrational potential are considered to be first-order in mechanical anharmonicity, quartic terms are second-order, and so forth. The notation (n, m) is used hereafter for the order of electrical (n) and mechanical (m) anharmonicity whereas the total order (n -I- m) is denoted by I, II,. Although our definition of orders is reasonable other choices are possible. Two key questions are (1) How important are anharmonicity contributions to vibrational NLO properties and (2) What is the convergence behavior of the double perturbation series in electrical and mechanical anharmonicity Both questions will be addressed later. Here we note that compact expressions, complete through order II in electrical plus mechanical anharmonicity, have been presented [19]. The formulas of order I contain either cubic force constants or second derivatives of the electrical properties with respect to the normal coordinates. Depending upon the level of calculation at least one order of numerical differentiation is ordinarily required to determine these anharmonicity parameters. For electrical properties, the additional normal coordinate derivative may be replaced by an electric field derivative using relations such as d p./dQidQj = —d E/dldAj.ACd, = —dk,/rjF where F is the field and k j is... 

The calculation of third and fourth derivatives is attractive for two reasons these are the quantities needed in the lowest-order treatment of vibrational anharmonicities, and a quartic surface is the simplest to exhibit a double minimum, i.e. the simplest model of a reaction surface. Moccia (1970) did apparently first consider the SCF third derivative problem. A detailed derivation of SCF and MCSCF third derivatives was given by Pulay (1983a) independently, Simons and J )rgensen (1983) also considered the calculation of MCSCF third, and even fourth, derivatives in a short note. As pointed out in Section II, third derivatives of the energy require only the first derivatives of the coefficients, and are thus computationally attractive. By contrast, fourth derivatives require the solution of the second-order CP MCSCF equations. The only computer implementation so far is that of Gaw etal.( 984) for closed shells, although the detailed theory has been worked out for the MCSCF case... 

Dunham (1932) was the first who understood the ideas and force of this method, and applied the WKBJ approach to the derivation of the second and third nonzero energy terms of the one-dimensional Schrddinger equation with the D-version of an anharmonic potential (see also Krieger et ah, 1967). Afterward and up to recent times, so many authors have applied the WKBJ method that if we attempt to enumerate them all, then quite half of the present article would be represented by the references alone. For the inquisitive reader we refer to some papers (Krieger et al., 1967 Kesarwani and Varshni, 1978, 1980 Zwaan, 1929 Kirschner and Watson, 1973 Stettler and Shatas, 1971 Froman and Froman, 1965 Kil-lingbeck, 1980 Hecht and Mayer, 1957) in which, without any difficulties, a whole body of literature devoted to the problem may be found i.e., by the principle of the Russian popular tale, which in our version becomes a reference, by reference, by reference, etc.,. . . , the reader will extract a turnip, i.e., a key to the essence. 

Finally we turn to the structured nature of the overtone bands. First we note that the structure in bands at low energies, due to dispersion effects, becomes washed out in their overtone spectra. In this respect the bands about 650 and 944 cm", assigned to components in the librational (2-<—0) and (3<—0) manifolds, are unusually sharp. They are sharper than the fundamental from which they derive and their widths are limited only by the spectrometer s resolution. These effects have been assigned to the impact of anharmonicity, since the librational well is more sinusoidal than quadratic in shape. Further analysis would address, mostly, band positions and this type of treatment can be followed in several texts [11]. We shall not pursue this avenue further except to draw attention to the distribution of intensities between the bands within any one overtone manifold. The intensities should be distributed according to the local density of states ( 2.6.2) of the symmetry species of the individual components in each manifold tables of these species have been published [12]. In this case the librational ground state species for a ion, imder T symmetry, is F2 and this gives rise to the (2<—0) overtone species A, E and F2. The distribution of intensity in the (3 —0) band will be according to A, F, 2F2. 

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