Hellmann-Feynman

This, the well-known Hellmann-Feynman theorem [128,129], can then be used for the calculation of the first derivatives. In nonnal situations, however, the use of an incomplete atom-centered (e.g., atomic orbital) basis set means that further terms, known as Pulay forces, must also be considered [130]. 

Note that the exact adiabatic functions are used on the right-hand side, which in practical calculations must be evaluated by the full derivative on the left of Eq. (24) rather than the Hellmann-Feynman forces. This forai has the advantage that the R dependence of the coefficients, c, does not have to be considered. Using the relationship Eq. (78) for the off-diagonal matrix elements of the right-hand side then leads directly to... 

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

Bala, R, Lesyng, B., McCammon, J.A. Extended Hellmann-Feynman theorem for non-stationary states and its application in quantum-classical molecular dynamics simulations. Chem. Phys. Lett. 219 (1994) 259-266. 

The intriguing point about the second set of equations is that q is now kept constant. Thus the vector ip evolves according to a time-dependent Schrddinger equation with time-independent Hamilton operator H[q) and the update of the classical momentum p is obtained by integrating the Hellmann-Feynman forces [3] acting on the classical particles along the computed ip t) (plus a constant update due to the purely classical force field). 

Here, the one-eleetron operator Zi (rj- Ra)/ ri-Rap is referred to as the Hellmann-Feynman foree operator it is the derivative of the Hamiltonian with respeet to displaeement of eenter-a in the x, y, or z direetion. 

The expressions given above for E(Z=0) and (dE/dZ)o ean onee again be used, but with the Hellmann-Feynman form for V. Onee again, for the MCSCF wavefunetion, the variational optimization of the energy gives... 

Another way of obtaining molecular properties is to use the Hellmann-Feynman theorem. This theorem states that the derivative of energy with respect to some property P is given by... 

This relationship is often used for computing electrostatic properties. Not all approximation methods obey the Hellmann-Feynman theorem. Only variational methods obey the Hellmann-Feynman theorem. Some of the variational methods that will be discussed in this book are denoted HF, MCSCF, Cl, and CC. 

These expressions are only correct for wave functions that obey the Hellmann-Feynman theorem. Flowever, these expressions have been used for other methods, where they serve as a reasonable approximation. Methods that rigorously obey the Flellmann-Feynman theorem are SCF, MCSCF, and Full CF The change in energy from nonlinear effects is due to a change in the electron density, which creates an induced dipole moment and, to a lesser extent, induced higher-order multipoles. 

Variational wave functions thus obey the Hellmann-Feynman theorem. 

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). 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

The first term is the Hellmann-Feynman force and the second is the wave function response. The latter now contains contributions both from a change in basis functions and MO coefficients. 

The central of these is recognized as the Hellmann-Feynman force. The two-electron... 

The central term is again the Hellmann-Feynman force, which vanishes since the two-electron operator g is independent of the nuclear positions. 

Figure 1 Convergence of the total energy and of the Hellmann-Feynman forces for ensembles of paramagnetic Fe atoms with 4 to 32 atoms. Part (a) shows the results of non-selfconsistent calculations performed with a fixed potential, part (b) the results of selfconsistent calculations. Full lines represent the RMM-DIIS (iterative diagonal-ization) results, broken lines the CGa (total-energy minimization) calculations. (4. text.

M. Menon and R. E. AUen, New technique for molecular-dynamics computer simulations Hellmann- Feynman theorem and subspace Hamiltonian approach , Phys. Rev. B33 7099 (1986) Simulations of atomic processes at semiconductor surfaces General method and chemisorption on GaAs(llO) , ibid B38 6196 (1988). 

The Hellmann-Feynman forces are also sensitive to the effect of moving ions on the basis set (pj) of the electronic wave function (, = This... 

We shall make use of the Hellmann-Feynman type formula ... 

Only the first one is a pure state mixing term. The Hellmann-Feynman expression of the two terms containing coefficient derivatives involves a bit of algebra. For K L... 

To obtain the Hellmann-Feynman theorem, we differentiate equation (3.67) with respect to X... 

Although the expectation value r )ni cannot be obtained from equation (6.70), it can be evaluated by regarding the azimuthal quantum number I as the parameter in the Hellmann-Feynman theorem (equation (3.71)). Thus, we have... 

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