SCF Energies

The gradient and second derivative components of the SCF energy can be expressed for both kinds of parametrization (see [28]) as... 

Use Mathcad to calculate the first approximation to the SCF energy of the helium atom... 

Using as many methods as are available to you for comparison (Mathcad, QBASIC, and TRUE BASIC), determine the self-consistent field (SCF) energies of the He atom and of the ions Li+, Be " ", and B +. Fill in the SCF column of Table 8-1. 

Procedure. To go from an STO-3G ealeulation to a CBS-4 ealeulation, simply replaee STO-3G with CBS-4 in the route seetion of the program used in Computer Projeet 8-1. Complete Table 8-2 by filling in the CBS-4 Energies of the atoms and ions listed in eolumns 1 and 3 of Table 8-2 and put them into eolumns 2 and 4 of the table. You will notiee that some of the simpler atoms (H through Be) do not have a listed CBS-4 Energies, but they do have an SCF energy, whieh should be used in its plaee. Caleulate the IP and eomplete eolumn 5. Pay speeial attention to spin niultiplieity and Hund s rule. The spin niultiplieity is rr + 1 where n is the number... 

An optimization of the transition structure geometry (yields the SCF energy). 

The optimization job converges quickly, computing an SCF energy of -113.69352 hartrees at the final point. The final structure is close to the starting molecule specification. 

Remember that the CBS models begin with a large enough SCF calculation to obtain the desired level of accuracy (see Chapter 7) therefore, no explicit extrapolation of the SCF energy is included. CBS extrapolation involves computing the second-order and infinite-order corrections to the energy. 

Because the interelectronic cusp is difficult to describe well with one-electron basis functions, pair correlation energies converge much more slowly (as N" ) than SCF energies (which converge as f ). This fact makes the use of CBS extrapolations of the correlation energy very beneficial in terms of both accuracy and computational cost. 

A iMuiliken population analysis foibw > ifre SCF energy results. This analysis partitions the charge on the molecule by atom. 

The only remaining question is the nature of the error function. Pulay suggested the difference FDS — SDF (S is the overlap matrix), which is related to the gradient of the SCF energy with respect to the MO coefficients. This has been found to work well in practice. 

Table 1. The 72-atom model examined by different theoretical methods. The energy differences (AE in kcal/mol) are calculated with respect to the lowest SCF energy. q(Fe) stands for Mulliken population charges on the Fe atoms q(S) and SS(b.i.) are the Mulliken population charges and the bond index for the bridging S atoms, respectively AEq is the calculated Mossbauer quadrupole splitting constant [mm/sec]. The PUHF spin states are those projected from the UHF wavefunction with 5 = 5,. 

Calculated as differences between total SCF energies of the neutral hydrocarbon and the corresponding positive ion radicals were calculated by the half-electron method, (34). Method of Longuet-Higgins and Pople eq. (90) was employed, (61). 

Absolute SCF energy - Absolute MP2 energy" G2 total energy - SCF Aifacid MP2 A addS G2 A acid expt6... 

Ed71- is obtained by a DEL keylist (a standard option of the NBO program) that deletes the possible 7t—7t interaction elements from the density matrix and recalculates the SCF energy in the absence of such interactions. For all model DEL calculations described in this section we employed the simpler RHF/6-31 + G level of theory. 

Most of the calculations have been done for Cu since it has the least number of electrons of the metals of interest. The clusters represent the Cu(100) surface and the positions of the metal atoms are fixed by bulk fee geometry. The adsorption site metal atom is usually treated with all its electrons while the rest are treated with one 4s electron and a pseudopotential for the core electrons. Higher z metals can be studied by using pseudopotentials for all the metals in the cluster. The adsorbed molecule is treated with all its electrons and the equilibrium positions are determined by minimizing the SCF energy. The positions of the adsorbate atoms are varied around the equilibrium position and SCF energies at several points are fitted to a potential surface to obtain the interatomic force constants and the vibrational frequency. 

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