Electron density Hellmann-Feynman theorem

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. 

The Hellmann-Feynman theorem demonstrates the central role of p, the electron density distribution, in understanding forces in molecules and therefore chemical bonding. The main appeal and usefulness of this important theorem is that it shows that the effective force acting on a nucleus in a molecule can be calculated by simple electrostatics once p is known. The theorem can be stated as follows ... 

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). 

Using Eq. (10.1) to obtain, with the help of the Hellmann-Feynman theorem [74], the derivative of AE at constant electron density p, taking the nuclear charge Z. as variable, namely... 

Beginning in the 1960s, Richard Bader initiated a systematic study of molecular electron density distributions and their relation to chemical bonding using the Hellmann-Feynman theorem.188 This work was made possible through a collaboration with the research group of Professors Mulliken and Roothaan at the University of Chicago, who made available their wave-functions for diatomic molecules, functions that approached the Hartree-Fock limit and were of unsurpassed accuracy. 

DFT has led to a substantial simplification of quantum-chemical computations. Like the Hellmann-Feynman theorem it expresses the reasonable assumption of a reciprocal relationship between potential energy and electron density in a molecule. In principle this relationship means that all ground-state molecular properties may be calculated from the ground-state electron density p(x, y, z), which is a function of only three coordinates, instead of a many-parameter molecular wave function in configuration space. The formal theorem behind DFT which defines the electronic energy as a functional of the density function provides no guidance on how to establish the density function p r) without resort to wave mechanics. 

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... 

If electronic densities, in particular, if density matrices of large molecules are available, then an important variant of the Hellmann-Feynman theorem can be used for the computation of forces acting upon various nuclei, to be used for macromolecular geometry optimization. 

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 ... 

The Hellmann-Feynman theorem holds for the density-functional expression for Eu, and it is instructive to work through this. Writing the one-electron eigenvalues in terms of the Hamiltonian h in (9), (11) becomes ... 

Derivation and integration of the above equation with respect to X will yield the difference of the energies of the real and noninteracting systems (E-Es) in the left hand side, and Es is exactly VS+TS) i.e., the sum of the potential energy plus the kinetic energy of a noninteractive system of electrons with density identical to the real system. Then using the Hellmann-Feynman theorem... 

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. 

However, a special form of the Hellmann-Feynman theorem, which can be expressed in terms of electronic density and does not require knowledge of the electronic wave function, has recently acquired additional significance. 

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. 

This chapter discusses theorems that are used in molecular quantum mechanics. Section 14.1 expresses the electron probability density in terms of the wave function. Section 14.2 shows how the dipole moment of a molecule is calculated from the wave function. Section 14.3 gives the procedure for calculating the Hartree-Fock wave function of a molecule. Sections 14.4 to 14.7 discuss the virial theorem and the Hellmann-Feynman theorem, which are helpful in understanding chemical bonding. 

For a bound stationary state, the generalized Hellmann-Feynman theorem is dE /dX = f tjt dH/d ) dT, where A is a parameter in the Hamiltonian. (In case of degeneracy, i/r must be a correct zeroth-order wave function for the perturbation of changing A.) 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. 

Use of the Hellmann-Feynman theorem in the evaluation (at constant electron density p) of the derivative... 

Following the Hellmann-Feynman theorem [49, 50] the gradient of energy is simply determined by the electron density p(f) ... 

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