Basin virial

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... 

The nuclear-electron terms in the operator on the right-hand side refer to the electronic coordinate Ti, which is excluded from the integration in eqn (6.78). The subscript T is not indicated on the coordinate appearing in p. The two-electron terms involve the coordinates Tj and f2 and integration of >j/ j/ over the coordinates of the remaining electronic coordinates and multiplication by the factor N N - l)/2, as indicated, yields the two-electron density matrix r (i i.< 2) (eqn (EL3)). Integration of the density in eqn (6.78) for the final electronic coordinate = r over the basin of the atom yields the basin virial T" (12) as indicated in eqn (6.79)... 

In addition to the basin virial i b(n) arising from the internal forces, which is identical to that obtained in the field-free case (eqn (8.192)), there are further contributions to the virials of the forces acting on the electrons in the basin of the atoms resulting from their interaction with the external fields. The surface virial reduces to the flux in the virial of the mechanical stresses... 

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. 

While the surface and basin virials are dependent upon the choice of origin for the vector r, it is possible to remove the origin dependence and... 

The surface virial for X, when added to the change in its basin virial vb(X), equals twice the change in its energy in forming the complex, and these changes in Vb(X) and FAX), are listed in Table 4. The surface virials,... 

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

The inherent valne of the topological method is that these atomic basins are defined by the electron density distribution of the molecule. No arbitrary assumptions are required. The atomic basins are quantum mechanically well-defined spaces, individnally satisfying the virial theorem. Properties of an atom defined by its atomic basin can be obtained by integration of the appropriate operator within the atomic basin. The molecular property is then simply the sum of the atomic properties. 

While the individual contributions to the virial tC ( 2) can be given physical interpretations, care must be exercised in this regard. The value of the total virial tC( 2) is independent of the choice of origin used in the definition of the virial operator, but this is not the case for the basin and surface terms treated separately. The origin-dependent terms appearing in the total virial for an atom are... 

According to eqn (6.103), the surface virial measures the disparity in the changes in the values of the electronic kinetic and potential energies over the basin of the atom required for the satisfaction of the virial theorem, as occasioned by the formation of chemical bonds. 

The separation of the virial into basin and surface contributions requires a choice of origin and a natural and physically useful choice is to place the origin of the position vector for atom A, the vector r, at the nucleus of the atom in question. If the vector from the nucleus of atom A to that of atom B is denoted by R b, then Rab = — rb nd, recalling that the surface normals are... 

Because L(Ci) vanishes for an atom, integration of eqn (7.4) over the basin of an atom yields the atomic virial theorem... 

The averaging of these commutators in the manner determined by the atomic action principle (eqn (8.225)) serves to define the atomic averages of the kinetic energy and of the virials of the internal and external forces acting on the basin of the atom, i.e. 

The first term is the virial of the force over the basin of the atom... 

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 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 atomic basins are quantum mechanically well defined and individually obey the virial theorem.Properties of the atoms can be obtained by integration of the proper operator within the atomic basin. The molecular... 

The topological method is strongly grounded in quantum mechanics. The atomic basins are well-defined quantum spaces, wherein the virial theorem holds. Integration of the elearon density within the basin does correspond to a physical observable. In principle, one can measure the electron density of a molecule (via an X-ray diffraction experiment) and integrate this experimental density within similarly defined atomic basins. Compared to all other population methods, the topological method rests on the firmest theoretical grounds and involves the fewest assumptions. 

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