Property second-order

For a first-order property and a fully variational wave function, we have used the Hellmann-Feynman theorem 

Here and for the remainder of the section we assume that the derivatives are evaluated at X = 0. 

The total energy of the system may be expanded in a Taylor series around X = 0, 

We know from time-independent nondegenerate perturbation theory that the energy may also be written as 

The energy correction to the vacuum is similar, but now 0) is an unoccupied state, accessible as a virmal state for the negative-energy electrons  

Second order properties derive from the second derivative term in Eq. [78]. For the most general case we should also allow for a second-order perturbation, yC ) — H 0) + (59f/aX)X + V2 from which differentiation 

Equation [86] is sometimes useful, e.g., when a knowledge of the excitation energies and polarized components of the transition moments for a few states could be used to estimate the a tensor from experiment. However, this sum-over-states formula is computationally about the worst approach to the theoretical evaluation of second-order quantities. To use it directly would require computing all the eccited states the finite basis permits and the appropriate transition moments. It can be solved indirealy, however.  

The superior approach is to simply compute the second derivative (Hessian) matrix analytically. Just as for force constants, this is the most accurate and efficient procedure when the second-derivative formulas have been programmed. Furthermore, any residual dependence on d V/dk is alleviated, just 

Whereas first-order properties can be obtained directly from the ground-state wave- 

Edmiston and K. Ruedenberg, Rev. Mod. Phys., 1963, 35, 457 H. Weinstein, R. Pauncz, and M. Cohen, Adv. Atomic Mol. Phys., 1971, 7, 97 K. Ruedenberg, in Modern Quantum Chemistry , (Istanbul Lectures), Part 1, Academic Press, New York, 1965, etc. 

As one might expect, the Frost model gives rather poor bond quadrupole moments owing to the diffuse nature of spherical gaussian charge distributions. Amos et al. have additionally extended the calculations to second-order bond properties.  

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Figures 4.14a and b are the analogs of Figs. 4.3a and b they schematicaUy describe second- and first-order transitions, respectively. It is the discontinuity in these second-order properties that characterizes a second-order transition.

The second-order property, the dipole polarizability, as given by the derivative formula eq. (10.31), is... 

We study three different approximations for removing unlinked diagrams in EOM-CC and show that these models provide second-order properties and transition probabilities that are close to those provided by CCLR in isolated molecular systems, but in a more convenient computational structure. 

Table 3 Second Order Properties Calculated by EOM-CCSD Methods...

Analogous considerations apply, for example, to the calculation of second-order properties, for which a very similar computational problem must be addressed. For typical applications this step constitutes 95-100% of the total computational effort, and a successful parallelization will therefore reflect directly on overall performance. 

As stated above, for a molecnle to demonstrate bnlk second order properties it is necessary for the dipole of all the molecnles to lie in the same direction. This situation is very difficnlt to attain in small molecnles where the tendency is for the molecules to crystallise with their dipoles opposing each other. This fact has led to a switch of interest to polymers that incorporate the small NLO molecnle. 

Recently, the assignment of the band at 980 cm to 28 has been doubted based on new calculations (this band is shifted to 976 cm if 28 is generated from 1,4-diiodobenzene (37), which is not unusual in the presence of iodine atoms. This shift may also be attributable to the change of the matrix host from argon to neon). ° On the other hand, ab initio calculations of the IR spectrum of 28 are complicated by the existence of orbital instabilities, the effect of which may (often) be negligible for first order properties (such as geometry and energy), but can result in severe deviations for second-order properties (vibrational frequencies, IR intensities). 

The time-independent second-order properties, such as the shielding constant or the nuclear spin-spin coupling constant (see Equations (2.8) and (2.9)) can be expressed by means of the perturbation theory as... 

The inclusion of correlation effects in the calculation of second-order properties, such as the polarizability, has been examined by Bartlett and co-workers, and an application to H2 reported.73 A very accurate Cl wavefunction due to Liu has been used to calculate the Compton profile for molecular H2, with results in good agreement with experiment.74... 

It should be noted that for one-electron properties, such as dipole moments, the [N/N] Pade approximants are invariant to changes of scale and shifts of origin in the reference spectrum8 whereas for second-order properties, such as polarizabilities, the [N/N+1] Pad6 approximants are to be preferred. Indeed, for polarizabilities the use of the form... 

We now proceed to consider second-order properties such as the polarizability and magnetizability tensors. Differentiating the first-order property Eq. 11, we obtain from the chain rule... 

We have established that, for a fully variational wave function, we may calculate the first-order properties from the zero-order response of the wave function (i.e., from the unperturbed wave function) and the second-order properties from the first-order response of the wave function. In general, the 2n -f 1 rule is obeyed For fully variational wave functions, the derivatives (i.e., responses) of the wave function to order n determine the derivatives of the energy to order 2n+ 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wave function to first order, but that the calculation of the energy to fourth order requires a knowledge of the wave-function response to second order. 

In the following, we shall first consider the optimization of C) and then go on to consider the evaluation of first- and second-order properties from the optimized wave function. 

Having set up the Hamiltonian, we may calculate the first- and second-order properties in the eigenvector representation. For the permanent electric and magnetic dipole moments, we obtain... 

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