Force Ehrenfest

The integration implied by dr in eqn (6,16) averages this force on the electron at r over the motions (i.e. positions) of all of the remaining particles in the system and the result is the force density F(r), the force exerted on the electron at r by the average distribution of the remaining particles in the total system. Integration of this force density over the basin of the atom 1 then yields the average electronic or Ehrenfest force exerted on the atom in the system. Even... 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

The individual contributions to the electronic potential energy of an atom are obtained by determining the action of the operator — r-V on V in the expression for the basin virial, eqn (6.19). The basin virial is the viried of the Ehrenfest force and the force density F(r) of eqn (6.16) is evaluated as the first... 

Averaging of this final operator expression for the virial of V in the manner indicated in eqn (6.69) for the potential energy density TFbWi which is the virial of the Ehrenfest force eqn (6.29), yields... 

Ehrenfest force acting on an atom in a molecule, /4(Q) = F(Q), is the vector sum of the forces exerted on it by each of its bonded neighbours, forces which vanish when the atom is free. The force exerted on atom Q by its bonded neighbour fj, the quantity FfnifT), is determined by the pressure a n acting on each of the surface elements dS(Q 0, r). 

Electrostatic potential maps have been used to make predictions similar to these (Scrocco and Tomasi 1978). Such maps, however, do not in general reveal the location of the sites of nucleophilic attack (Politzer et al. 1982), as the maps are determined by only the classical part of the potential. The local virial theorem, eqn (7.4), determines the sign of the Laplacian of the charge density. The potential energy density -f (r) (eqn (6.30)) appearing in eqn (7.4) involves the full quantum potential. It contains the virial of the Ehrenfest force (eqn (6.29)), the force exerted on the electronic charge at a point in space (eqns (6.16) and (6.17)). The classical electrostatic force is one component of this total force. 

The statement of the atomic action principle given in eqn (8.145) is a variational principle which enables one to derive the properties of an atom in a molecule—it is an atomic variation principle. We shall use it first to derive the atomic statement of the Ehrenfest force law, the equation of motion of an atom in a molecule. This is accomplished through a variation of fl]... 

The atomic statement of the Ehrenfest force law is obtained by setting the generator F equal to the momentum a. The commutator of the many-particle Hamiltonian eqn (8.226) and a can be expressed as... 

Ehrenfest force density, eqn (6.16) kinetic energy density, eqn (5.49)... 

The expectation value of the commutator for the virial operator G(r) = r p yields 2T(Q) + vb( 2), twice the atom s electronic kinetic energy, T(Q), together with the virial of the Ehrenfest force exerted over the basin of the atom, vfc( 2) [4], In a stationary state these contributions are balanced by v,(S2), the virial of the Ehrenfest force acting over the surface of the atom. Expressing by v( 2), the total virial for atom 2, the virial theorem for a stationary state may be stated as [4]... 

The virials of the Ehrenfest force exerted over the basin and the surface of the atom, with the origin for the coordinate r placed at the nucleus of atom 2, are given respectively in Equations (8) and (9)... 

The virial theorem plays a dominant role in the definition of pressure, in both classical and quantum mechanics. The following section demonstrates that the pv product for a proper open system is proportional to the surface virial, Equation (9), the virial of the Ehrenfest forces exerted by the surroundings on the open system [9,12],... 

Equation (18) is identical in form and content to the atomic statement of the virial theorem Equation (7) - the virial theorem for a proper open system - with the petit virial Zp being the analogue of the basin virial Vb and the surface flux virial Zs/ the analogue of the surface virial vs, the virial of the Ehrenfest forces acting on the surface of the open system. 

The total virial for an open system v(Q), is of course, origin independent since it equals —27 ( 2). This is not the case, however, for its expression in terms of the sum of its basin v ( 2) and surface v,v( 2) contributions, whose values, according to their definitions in Equations (8) and (9), are dependent up a choice of origin. Consider a coordinate transformation denoted by r = r + 5R caused by a shift SR in the origin. This has the effect of changing the basin and surface virials by the same absolute amount, equal to the virial of the Ehrenfest force as given in Equation (23),... 

The Ehrenfest force acting on an atom may be similarly expressed in terms a sum of interatomic contributions, one from the force exerted by each of its bonded neighbours... 

The interatomic surface virial vs(A B) vanishes for the separated atoms and hence vs(A B) is the contribution to the energy of formation of the molecule arising from the creation of an interatomic surface S(A fi) between atoms A and B, a most useful result. The Ehrenfest force F(A B) is attractive when the force exerted on the density of atom A is directed at B, the situation found for all bonded interactions. The interatomic surface virial vs (A B) is negative in such a case and the formation of the surface S(A Z ) contributes to the stability of the system. 

While Pendas makes no attempt to disprove the quantum definition of pressure obtained through the scaling procedure of Marc and McMillan [22] as presented here, he does state that "the use of electron-only scaling to study stressed situations, where the virials due to the nuclear system cannot be neglected, is not a very consistent procedure." In reality, the physics of an open system does not neglect the virials due to the nuclear system, they are included in the total virial that is defined by taking the virial of the Ehrenfest force, see for example Equation (13). The virial theorem illustrates... 

Another important result governing the mechanics of an atom in a molecule is obtained from equation 16 when the operator F is set equal to the momentum for an electron. The result in this case is an expression for the force acting on the electrons in an atom, the Ehrenfest force, a force not to be confused with the Hellmann-Feynmann force acting on a nucleus. The expression for the Ehrenfest force is equivalent to having Newton s equation of motion for an atom in a molecule, as it determines all of the mechanical properties of the atom. The force F Q) is determined entirely by the pressure acting on the surface of the atom [Pg.44]

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