Property first-order

Properties of interest can be static or dynamic (i.e., time or frequency dependent) first, second, or higher order and one- or two-particle type. First-order static one-electron properties (dipole and quadrupole moments, electric 

Another, more general way to define a property is via perturbation theory, where the usual electronic Hamiltonian, H (now H(0)) has a perturbation X0 added to it. (The 0 argument of H should be taken to mean that the Hamiltonian does not contain the perturbation of interest. It should not be confused with a Hamiltonian that ignores elearon correlation.) Then 

Inserting Eqs. [77] and [78] into Eq. [76], and insisting that the coefficient of each power of X be the same on both sides of the equation, we have 

Provided we know the solution to the unperturbed Schrodinger equation 

The SCF solution for a first-order, one-electron property is usually argued to be relatively accurate. The reason follows from the Moller-Plesset theorem. Earlier we found that the first correlation correction to Oq comes solely from double excitations, 0 + However, ( I o 0 Ii / ) = 0 

First-order properties can be written as the expectation value of the appropriate operator as follows  

The first-order property for which most experimentaF - and theoretical results exist is the (electric) dipole moment. The literature on calculations of dipole moment is almost as large as that on wavefunctions. 

If P were known exactly, the value of a first-order property calculated from equation (12) would be exact. In practice, only an approximation to P is known, and it is important to know how the expectation value differs from the exact value. Since errors in calculated dipole moments due to the breakdown of the Bom-Oppenheimer approximation are likely to be small (typically 0.002 a.u.), and for most molecules relativistic effects can be ignored, there are two separate remaining problems in practice. The first concerns the likely accuracy when the wavefunction is at the Hartree-Fock limit, the second the effect of using a truncated basis set to obtain a wavefunction away from the Hartree-Fock limit. 

Gordy and R. L. Cook, Microwave Molecular Spectra , Wiley, New York, 1970. 

McClellan, Tables of Experimental Dipole Moments , Freeman, San Francisco, 1963. R. D. Nelson, D. R. Lide, jun., and A. A. Maryott, Selected Values of Electric Dipole Moments for Molecules in the Gas Phase , Nat. Bur. Standards (US) NSR DS-NBSIO Govt. Printing Office, Washington D.C., 1967. 

McClellan, Tables of Experimental Dipole Moments , Freeman, San Francisco, 1963. 

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In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

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). 

Table 3 Calculated first-order properties for HCN using five basis sets which gave the same energy to five decimal places. Atomic units are used throughout...

From Table 3 it can be seen that as a rough rule, if the energy is determined to n decimal places, first-order properties are stable to n[2. There is one other rather interesting result of these calculations not shown in the Table. For all properties other than F and FG the lowest values were those from the fully self-consistent wavefunctions which gave the lowest energy. As upper bound principles do not exist for other operators, this must be a coincidence. 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

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

A further extension of the PCM basic model to treat excited states considers the calculation of their properties. A basic result of the QM shows that first order properties of isolated molecules can be expressed as derivatives of the energy with respect to a proper perturbation. The same statement can be generalized to molecules in solution, considering the derivatives of the basic energetic functional (Gk.)- Within this scheme, the properties of excited states may be evaluated for both the QM approaches described in the previous section, i.e., SS and LR. Here, we will focus on the evaluation of the properties by using the PCM-LR approach. 

The first-order properties of excited states The effect... 

The knowledge of P4 permits, in turns, to evaluate the changes upon excitation of the first-order properties. For the most common example of the electric dipole moment, its variation between the excited and the ground state is given as ... 

Let us consider the first-order properties for the optimized variational energy E (x) in Eq. 8. Using the chain rule, we obtain... 

Combining Eq. 9 and Eq. 10, we obtain the following simple expression for first-order properties (e.g., for the permanent electric and magnetic dipole moments) for a fully variational wave function ... 

In short, to calculate first-order properties for a fully variational wave function, we need not evaluate the response of the wave function dX/dx. This is an extremely important result, which forms the basis for all computational techniques developed for the evaluation of molecular gradients (as well as for all higher-order properties). 

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... 

Let us summarize our results so far. We have found that the first-order properties may be calculated according to the expression... 

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. 

The molecular properties may now be calculated from this Lagrangian as for any fully variational energy [1]. In particular, the first-order properties are obtained as... 

There have been several previous ab initio molecular orbital studies of oxirane using the experimental geometry.30.3it34,35,38,39,63) These have been mostly concerned with ionization potentials and first order properties such as multipole moments. Some details are given in Table 10. As with aziridine, all the theoretical dipole moments except those calculated with the minimal Slater basis 33) are higher than the experimental value. 

In such cases the expression from first-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, ME or CC), the Hellmann-Feymnan theorem does not hold, and a first-order property calculated as an... 

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