Virial ratio

The virial ratio is, as we noted above, 1.3366 for the separate-atom AO basis MO calculation, i.e. not 1.0. Now within the confines of the linear variation method (the usual LCAO approach) there is no remaining degree of freedom to use in order to constrain the virial ratio to its formally correct value (or indeed to impose any other constraint). Thus imposing the correct virial ratio on the linear variation method is, in this case, not possible without simultaneously destroying the symmetry of the wave function. Only by optimising the non-linear parameters can we improve the virial ratio as the above results show. Even at this most elementary level, the imposition of various formally correct constraints on the wave function is seem to generate contradictions. 

It is of interest to compare the data in Table 2 with the results of the investigation by KPTT. Only the tabulated function with nonzero y is qualitatively comparable with the KPTT wavefunction for He these two wavefunctions have about the same energy (that of KPTT is — 2.9000). Our wavefunction has die exact virial ratio ( — potential energy/kinetic energy=2 vs. the KPTT value of 2.074) this may be interpreted as an indication that it is at a better overall scaling than the KPTT function. That function, however, gives better results than ours for (pi P2) KPTT, 0.1545 vs. our 0.1928 and the exact value of 0.1591. 

As a consequence of Eq. (6.27), the sign of the Laplacian at a point determines whether the negative potential energy or the positive kinetic energy is in excess of the virial ratio po, / kin = 2 at that point. In negative regions of the Laplacian... 

To compare the kinetic and potential energy densities on an equal standing, instead of the 2 1 virial ratio, Cremer and Kraka [110] evaluate the total electronic energy density at the BCP ... 

One of the most interesting predictions of the new method is the issue of the Pauli exclusion principle. When one attempts to calculate the energy of an antipair one immediately discovers the system is not bounded the Virial ratio does not work. More work is needed, but once continuum electrons are brought into the structure, it should be possible to directly calculate the energy deficit encountered when the Pauli exclusion principle is violated. This gives a meaningful explanation of the principle in terms of constructive and destructive interference. 

All theses 7 parameters were obtained from squared-error minimizations. The data analysis reveals that the best fits for hydrogen follow a very shallow minimum valleys so that the reported 7 parameters obtained from these fits should be taken with a grain of salt. The value 7j = 2 which was tested here is consistent with - and loosely justified by - the virial ratio for H, V e/E = 2. A reasonable explanation for possible distortions of the 7 value is perhaps linked to the basis used for hydrogen. This point is briefly examined with the help of DFT results (Table 5) and SCF computations using enriched bases. 

The student is then shown that a plot of E vs. gives a minimum for 4 = 1 for the H atom. In other words, we get them to appreciate the variation principle from this simple optimization of 4. The virial ratio (—V/T) also is introduced as an important concept of electronic behavior. We do not discuss the excited states of the H atom, but restrict ourselves solely to the ground state. The only connection with experiment is the ionization energy of the H atom and other one-electron atoms. The student is told that orbital energy and total energy mean one and the same thing for the H atom, but that as soon as we turn to He or more complicated atoms, the situation will be dramatically different. Constant contours of the wavefunction are introduced. 

We have developed a software application, ATOMPLUS,i38 which can be used in a turnkey fashion to illustrate theoretical features of two electron atoms, such as basis set effects, correlation energy, virial ratio, variational principle, and multideterminantal wavefunction. The program is provided in executable format for either the IBM or Macintosh platform. The source code can be obtained from Project Seraphim and, since it is written in FORTRAN 77, it compiles readily on other platforms. 

Virial Ratio = -/= 0.985374 Reference State Orbitals for 3 Filled Orbitals by Column... 

In conclusion, CSB originates from the equilibrium conditirai of the bond, defined by the virial ratio. It is promoted by two main factors ... 

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