First anharmonicities

The first term is the harmonic expression. The next is termed the first anharmonicity it (usually) produces a negative contribution to E(vj) that varies as (vj + 1/2)2. The spacings between successive Vj —> Vj + 1 energy levels is then given by ... 

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. 

The first anharmonicity [Eq. (69)] contains four terms. Using Eq. (40) the first term may be written as... 

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... 

The Cl first anharmonicity contains three terms [Eq. (118)], the first of which... 

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). 

In MCSCF calculations the orbital rotational parameters are contained in the variational space, and the solution of the response equations through order n then determines the energy through order 2n + 1. For example, the first-order response equations are sufficient to determine the Hessian as well as the first anharmonicity. 

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. 

The vibrational anharmonicity gives a first additional term in the vibrational energy of — xdtfv + he, [The second term is the coefficient of (v + i) and is shown by observed spectra to be negligibly small.] Here, Xe is the (first) anharmonicity constant and is the equilibrium wave-number, for small displacements. The energy of the vth vibrationally excited state is then given by... 

The Morse oscillator allows the first anharmonic correction to be related directly to the curvature and depth of the potential well ... 

The first anharmonicity constant, at the first overtone (V02), was computed by using X = voi — (vo2/2), whereas the anharmonicity constant at the second overtone (fos) was obtained from X = (vo2/2) — (vqj/S). Fundamentals, overtones and anharmonic constants are expressed in cm units. In the case of cycloalkanes (30) and hydrocarbons (31), a Birge-Sponer plot was used to determine anharmonicity constants. Axial and equal stand for axial and equatorial CH groups of cycloalkanes, respectively. 

Let us consider further reasons of pol5rmer chains breaking at so small stresses, which can be on order lower than ftacture macroscopic stress (i.e., at h5rpothetical k = 0.1). The reasons were pointed for the first time in Refs. [1, 26]. Firstly, anharmonicity intensification in fracture center gives the effect, identical to mechanical overloading effect [26]. Quantitatively this effect is expressed by the ratio of thermal expansion coefficient in fracture center and modal thermal expansion coefficient [5]. The second reason is close inter communication of local yielding and fracture processes [ 1]. This allows to identify fracture center for nonoriented polymers as local plasticity zone [27, 28]. The ratio uJ(X in this case can be reached -100 [5]. This effect compensates completely k reduction lower than one. So, for PC ala 70, K- = 0.44, a. = O.IE. 700 MPa and fiien o = o a /K,a 23 MPa, that by order of magnitude corresponds to experimental value Oj. for PC, which is equal approximately to 50 MPa at T= 293 K [7]. 

The vibrational/rotational energy states of a diatomic molecule can now be written to include not only the RRHO approximation but also in terms of the correction factors including the first anharmonicity correction, centrifugal distortion, and vibration-rotation coupling (Section 6.3 - 6.5). 

Figure 6-8. The contribution of the first anharmonicity, centrifugal distortion, and rotation-vibration coupling for H Cl vibration/rotation energy levels relative to the energy values computed from the rigid rotor harmonic oscillator approximation (RRHO). The numbers in parenthesis correspond to the contribution of each correction term. The constants were obtained from Table 6-2.

The constants c, that arc generated by the expansion of Equation 6-60 can be assigned to the various physical constants such as the first anharmonicity, centrifugal distortion, vibration-rotation coupling, and so on. 

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