Hellmann-Feynman electrostatic theorem

When s is a nuclear coordinate X , eqn (5.36) gives the Hellmann-Feynman electrostatic theorem... 

In order to perform ah initio molecular dynamics in excited states, the forces on the atoms are computed on the fly. There are two different implementations of force calculations for the two TDDFT schemes, P-TDDFT and LR-TDDFT. In the first case, the excited state is obtained with the promotion of one electron from the highest occupied molecular orbital (HOMO) to a selected unoccupied molecular orbital (the lowest one, LUMO, in our case). The corresponding KS excited single-determinant configuration is taken for the computation of the electronic density, which is then used to compute the forces on the atoms according to the Hellmann-Feynman electrostatic theorem [35]... 

Thus the effective force acting on a nucleus in a molecule can be calculated by simple electrostatics as the sum of the Coulombic forces exerted by the other nuclei and by a hypothetical electron cloud whose charge density ep(x, y, z) is found by solving the electronic Schrodinger equation. This is the Hellmann-Feynman electrostatic theorem. The electron probability density depends on the parameters defining the nuclear configuration p = p x, y, z x , y , Zc x, ...). 

For further applications of the Hellmann-Feynman electrostatic theorem, see B. M. Deb, ed.. The Force Concept in Cherrustry, an Nostrand Reinhold, 1981. 

For a bound stationary state, the generalized Hellmann-Feynjiaan theorem is dEJdX =S ipn dH/dX) dr, where A is a parameter in the Hamiltonian. Taking A as a nuclear coordinate, we are led to the Hellmann-Feynman electrostatic theorem, which states that the force on a nucleus in a molecule is the sum of the electrostatic forces exerted by the other nuclei and the electron charge density. 

In order to construct a numerically useful MD force field or a corresponding potential energy for a molecular system, one has to map the quantum mechanical interactions onto a reasonably simple classical expression. Here, the Hellmann-Feynman electrostatic theorem is useful [10] ... 

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. 

Assuming the validity of the Born-Oppenheimer approximation, the electrostatic Hellmann-Feynman (H-F) theorem expresses the force F A on a nucleus A, of charge ZA, in a molecule or solid, as... 

The electrostatic Hellmann-Feynman theorem states that for an exact electron wave function, and also of the Hartree-Fock wave function, the total quantum-mechanical force on an atomic nucleus is the same as that exerted classically by the electron density and the other nuclei in the system (Feynman 1939, Levine 1983). The theorem thus implies that the forces on the nuclei are fully determined once the charge distribution is known. As the forces on the nuclei must vanish for a nuclear configuration which is in equilibrium, a constraint may be introduced in the X-ray refinement procedure to ensure that the Hellmann-Feynman force balance is obeyed (Schwarzenbach and Lewis 1982). 

Hirshfeld (1984) found the electrostatic charge balance at the F nuclei, based on the experimental deformation density, to be several times more repulsive (i.e., anti-bonding) than that of the promolecule. Very sharp dipolar functions at the exocyclic C, N, and F atoms, oriented along the local bonds, were introduced in a new refinement in which the coefficients of the sharp functions were constrained to satisfy the electrostatic Hellmann-Feynman theorem (chapter 4). The electrostatic imbalance was corrected with negligible changes in the other parameters of the structure. The model deformation maps were virtually unaffected, except for the innermost contour around the nuclear sites. 

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... 

The electrostatic Hellmann-Feynman theorem is a special form of the general Hellmann-Feynman theorem. This form of the theorem can be expressed in terms of electronic density, and no explicit form of the electronic wave function is needed. The electrostatic Hellmann-Feynman theorem is of special significance in view of new developments in the construction of macromolecular electron densities and density matrices without using wave functions. ... 

An important application of the electrostatic Hellmann-Feynman theorem within the AFDF framework is the basis of a novel, macromolecular geometry optimization technique. Assume that the electronic density p(r) of a molecule of N nuclei and k electrons is available. The components of the position vector of nucleus a and those of position vector r, of electron i are denoted by X, Y, and Z ... 

This approach is advantageous if MEDLA, ALDA, or ADMA macro-molecular electronic densities are available. By carrying out a simple three-dimensional integration in the first term and a trivial summation in the second term of Eq. (315), the electrostatic Hellmann-Feynman theorem can be used for the computation of forces acting on the nuclei of macromolecules. 

In the preceding sections we have studied diatomic interactions via U(R). However, the study of diatomic interactions can also be carried out in terms of the force F(R) instead of the energy U(R), where R denotes the internuclear separation. Though there are several methods for the calculation of the force, the electrostatic theorem of Hellmann (1937) and Feynman (1939) is of particular interest in this section, since the theorem provides a simple and pictorial method for the analysis and interpretation of interatomic interactions based on the three-dimensional distribution of the electron density p(r). An important property of the Hellmann-Feyn-man (HF) theorem is that underlying concepts are common to both the exact and approximate electron densities (Epstein et al., 1967, and references therein). The force analysis of diatomic interactions is a useful semiclassical and therefore intuitively clear approach. And this results in the analysis of diatomic interactions via force functions instead of potential ones (Clinton and Hamilton, 1960 Goodisman, 1963). At the same time, in the authors opinion, it serves as a powerful additional instrument to reexamine model diatomic potential functions. 

From the chemical point of view, we must say these equations are not tractable and provide no useful information. In common, the study carried out by many authors (Salem, 1963b Byers-Brown, 1958 Byers-Brown and Steiner, 1962 Bader, 1960b Murrell, 1960 Berlin, 1951 Ben-ston and Kirtman, 1966 Davidson, 1962 Benston, 1966 Bader and Bandrauk, 1968b Kern and Karplus, 1964 Cade et al., 1966 Clinton, 1960 Phillipson, 1963 Empedocles, 1967 Schwendeman, 1966) on the force constants is based on the application of the virial and the Hellmann-Feynman or the electrostatic theorems. In particular, the Hellmann-Feynman theorem provides the expression for ki which relates the harmonic force constant to the properties of molecular charge distribution p(r), i.e., it follows (Salem, 1963b) that... 

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.,... 

If reasonably accurate electronic densities are available, then the forces acting on the nuclei can be approximately determined by a simple application of the electrostatic theorem, an important variant of the Hellmann-Feynman theorem. In turn, these forces can be used for geometry optimization. 

The electrostatic theorem, as a special case of the Hellmann-Feynman theorem, has some limitations. These limitations are briefly reviewed helow, following the discussion of the connections between the Hellmann-Feynman theorem and general properties of potential energy hypersurfaces. ... 

Although not yet obvious from the deceptively simple form of equation (37), this equation, the electrostatic Hellmann-Feynman theorem, allows one to use the electronic density and the simple internuclear Coulomb interactions to describe the forces acting on the nuclei of the molecule. A simple, classical interpretation of this theorem provides the key to the use of macromolecular electronic densities, such as those obtained within the MEDLA, ALDA, or ADMA methods, for the computation of forces within the macromolecule. 

The Somoyai function is defined in terms of the electronic density function and the composite nuclear potential, providing a 3D shape representation of the bonding pattern within the molecule under study. Some of the topological techniques of molecular shape analysis have been reviewed, with special emphasis on applications to the Somoyai function. A combination of a family of recently introduced ab initio quality macromolecular electronic density computation methods with the electrostatic Hellmann-Feynman theorem provides a new technique for the computation of forces acting on the nuclei of large molecules. This method of force computation offers a new approach to macromolecular geometry optimization. 

The Hellmann-Feynman theorem states that the forces on the nuclei in a molecule can be calculated according to the classical electrostatic interaction formula... 

Eqs. (3.9) and (3.10) are, then, the mathematical statements of the Hellmann-Feynman theorem, i. e. once the wave functions are known, the forces on the nuclei can be calculated according to the principles of electrostatics. This Helhnann-Fe5mman theorem is very important for the calculation of force constants of polyatomic molecules, and can also be used to develop an electrostatic model of chemical bonds [27—32). 

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