Sum-over-states

Note that equation (A3.11.1881 includes a quantum mechanical trace, which implies a sum over states. The states used for this evaluation are arbitrary as long as they form a complete set and many choices have been considered in recent work. Much of this work has been based on wavepackets [46] or grid point basis frmctions [47]. 

A second method is to use a perturbation theory expansion. This is formulated as a sum-over-states algorithm (SOS). This can be done for correlated wave functions and has only a modest CPU time requirement. The random-phase approximation is a time-dependent extension of this method. 

SOS (sum over states) an algorithm that averages the contributions of various states of the molecule... 

The tensor is given in lower-triangular format (i.e. a. standard orientation. The Approx polarizability line gives the results of the cruder polarizability estimate using sum-over-states perturbation theory, which is suggested by some older texts. 

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

The basic problem of statistical mechanics is to evaluate the sum-over-states in equation 7.2 and obtain Z and F as functions of T and any other variables (such as external magnetic fields) that might appear in %. Any thermodynamic observable of interest can then be obtained in a straightforward manner from equation 7.5. In practice, however, the sum-over-states often turns out to be prohibitively difficult to evaluate. Instead, the physical system is usually replaced with a simpler model system and/or some simplifying approximations are made so that the sum-over-states can be evaluated directly. 

Once HsoS and Hs h are both specified, the basic problem is to once again find a way to compute the sum-over-states appearing in the expression defining the partition function (equation 7.2) ... 

A sum-over-states expression for the coefficient A for the expansion of the diagonal components faaaa was derived by Bishop and De Kee [20] and calculations were reported for the atoms H and He. However, the usual approach to calculate dispersion coefficients for many-electron systems by means of ab initio response methods is still to extract these coefficients from a polynomial fit to pointwise calculated frequency-dependent hyperpolarizabiiities. Despite the inefficiency and the numerical difficulties of such an approach [16,21], no ab initio implementation has yet been reported for analytic dispersion coefficients for frequency-dependent second hyperpolarizabiiities which is applicable to many-electron systems. 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

Malkin, V. G., Malkina, 0. L., Casida, M. E., Salahub, D. R., 1994, Nuclear Magnetic Resonance Shielding Tensors Calculated With a Sum-Over-States Density Functional Perturbation Theory , J. Am. Chem. Soc., 116, 5898. 

J(PQ) and J(PN)). Trends in J(P170) have been used to study the protonation of phosphoryl tribromide, phosphoryl trifluoride and dif1uorophosphoric acid. Nitric acid had a greater protonation power than Hammett acidity functions indicated.77 This coupling has been the subject of a semiempirical sum over states theoretical study.7 ... 

The high value of the electron density at the nucleus leads to the enhancement of the electron EDM in heavy atoms. The other possible source of the enhancement is the presence of small energy denominators in the sum over states in the first term of Eq.(29). In particular, this takes place when (Eo — En) is of the order of the molecular rotational constant. (It is imperative that a nonperturbative treatment be invoked when the Stark matrix element e z(v /0 z v / ) is comparable to the energy denominator (Eq En) [33].) Neglecting the second term of the right-hand side of Eq.(29), which does not contain this enhancement factor [8, 27], we get... 

Meyers F, Marder SR, Pierce BM, Bredas JL (1994) Electric field modulated nonlinear optical properties of donor-acceptor polyenes sum-over-states investigation of the relationship between molecular polarizabilities (a, p, and y ) and bond length alternation. J Am Chem Soc 116 10703-10714... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

The function Q is the canonical partition function or sum over states. The eigenvalues En are difficult, if not impossible, to obtain. It is significant,... 

A possible computational strategy is to calculate Xs(r O first using the standard sum-over states formula (Equation 24.80). Equation 24.75 can be used next to generate successive Born approximations of the functions f (r). For instance, the first Bom approximation would be... 

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