Hellmann-Feynman theory

Given an initial form for the potential, we use the Marquardt algorithm (93,94) to minimize 2- This algorithm requires the evaluation of terms such as dE k/dFlm . Such derivatives can be readily calculated using Hellmann-Feynman theory, which states that... 

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). 

By defining all these quantities as explicit functions of A, we can relate the density functional quantities to those more familiar from quantum chemistry. The exchange-correlation energy of density functional theory can be shown, via the Hellmann-Feynman theorem [38, 37], to be given by a coupling-constant average, i.e.. 

Equations (3.20) and (3.21) represent an identity in Hartree-Fock theory. (The Hellmann-Feynman and virial theorems are satisfied by Hartree-Fock wavefunc-tions.) The particular interest offered by (3.21) lies in the fact that 7 = 1 appears to be the characteristic homogeneity of both Thomas-Fermi [62,75,76] and local density functional theory [77], in which case (3.20) gives the Ruedenberg approximation [78], E = v,e,-, while (3.21) gives the Politzer formula [79], E = Vne-... 

Before turning to many-electron molecules, it is useful to ask Where does the energy of the chemical bond come from In VB theory it appears to be connected with exchange of electrons between different atoms but in MO theory it is associated with delocalization of the MOs. In fact, the Hellmann-Feynman theorem (see, for example, Ch.5 of Ref.[7]) shows that the forces which hold the nuclei together in a molecule (defined in terms of the derivatives of the total electronic energy with respect to nuclear displacement) can be calculated by classical electrostatics, provided the electron distribution is represented as an electron density P(r) (number of electrons per unit volume at point r) derived from the Schrodinger wavefunction k. This density is defined (using x to stand for both space and spin variables r, s, respectively) by... 

It should be pointed out that Schwarz (20),using double perturbation theory,has demonstrated that it is possible to rationalize the relativistic bond length contraction in terms of the attractive Hellmann-Feynman force due to the relativistic change in electron density.In such an approach it would be necessary to analyze and get a physical picture of the relevant density changes... 

Our primary goal was the simulation of entire atomic systems, thus made of electrons and nuclei. As mentioned earlier (see Sect. 2.3), in a large class of systems (e.g. not too high temperature) one can decouple the motion of nuclei and electrons within the Born-Oppenheimer approximation. The previous section was then devoted to the Density Functional Theory solution of the electronic structure problem at fixed ionic positions. By computing the Hellmann-Feynman forces (11) we can now propagate the dynamics of an ensemble of (classical) nuclei as described in Sect. 2.3, using e.g. the velocity verlet algorithm [117]. 

The exchange-correlation hole is of considerable interest in density functional theory, as the exact exchange-correlation energy may be expressed in terms of this hole. By use of the Hellmann-Feynman theorem, one may write the exchange-correlation energy as the electrostatic interaction between the density and the hole, averaged over coupling constant[13], i.e.,... 

The second possibility is to use a gradient code, if this is available for the chosen method. The third method is the simplest, namely to evaluate E2 as an expectation value. This method is equivalent to the other two, if Eq satisfies a stationarity condition, like the Brillouin condition of Hartree-Fock theory. For non-stationary approaches, like MP2 or CC, the methods based on differentiations of the expectation value are more reliable. This is related to the validity or non-validity of the Hellmann-Feynman theorem. 

This article extends the Hellmann-Feynman amalysis and the range of application of the related technique of the coupling constant integration, to studying the exchange-correlation density functionals in the subsystem resolution. In particular the nonadditive functionals have been examined in a more detail, with a special emphasis placed upon the density functional for the nonadditive kinetic energy of the noninteracting system, which appeares in the Kohn-Sham theory for subsystems. 

Pack and Hirschfelder (133) discuss the energy corrections to the Bom-Oppenheimer approximation, while Bunker (134) discusses the partial breakdown of the Bom-Oppenheimer approximation. The Hell-mann-Feynman theorem is also discussed by Tuan (135) with reference to multiconfiguration SCF-theory, while Loeb and Rasiel (136) discuss constraints upon the LiH molecule with reference to the Hellmann-Fe5mman theorem. King (137) developed a theory of effective Cartesian force constants which relate to the sum of the squares of the normal frequencies these effective force constants are independent of the inter-nuclear repulsion of the nuclei in a molecule, and thus removes one of the indeterminacy of the Hellmann-Feynman method described above. 

In such a case, structure optimization is mandatory and may be introduced by calculating the interatomic forces using electronic-structure theory the alert theorist always checks the starting geometry. If the electronic structure of the molecule or solid and its wave function Y are known, the famous Hellmann-Feynman theorem [3,241] gives access to the force F between two nuclei separated by a distance R... 

Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

Inconsistencies in the theory such as the ones discussed above may be considered connected with the lack of preservation of certain formal equivalencies in an approximate theory. An obvious relation that follows from the Hellmann-Feynman theorem is the expression for the bond order as a derivative of the total energy... 

The first-order perturbation theory result Eq. (9.22) is a special case of the Hellmann-Feynman theorem (Problem 14.14). 

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