Hartree-Fock MO calculations

Yamabe et al. (1980) investigated this problem in more detail and confirmed the assumption that the thermal and photolytic dediazoniation along the path of least motion (a Ciy dissociation) is unfavorable. Nevertheless, it is possible if the symmetry of the system is reduced to C2 or Q. Closed-shell Hartree-Fock MO calculations led to the result that the lowest excited state of diazomethane can initiate the dediazoniation through the bent-in-plane path (C5.). These results were corroborated by Csizmadia s group (Wang et al., 1991) in ab initio calculations with 3-21G and... 

MO calculations for the azide group have involved a variety of approaches and degrees of sophistication. Probably the most reliable wave functions are obtained from recent self-consistent-field Hartree-Fock MO calculations. These are reviewed here. For a discussion of other calculations [10-17], reference is made to the excellent review by Treinin [18]. 

For the benzenoid aromatic molecules considered here, we wish to include electron correlation for the N electrons of the 7t-eleetron system but not for the 2n a electrons. This is achieved most readily if we first carry out a standard Hartree-Fock MO theory calculation, yielding a set of orthogonal self-consistent field (SCF) molecular orbitals tp(. The spin-coupled wavefunction may then be written... 

Summary of the steps in a single-point Hartree-Fock (SCF) calculation using the Roothaan-Hall LCAO expansion of the MO s... 

As we just saw, MP2 calculations utilize the Hartree-Fock MOs (their coefficients c and energies e). The HF method gives the best occupied MOs obtainable from a given basis set and a one-determinant total wavefunction i(i, but it does not optimize the virtual orbitals (after all, in the HF procedure we start with a determinant consisting of only the occupied MOs - Sections 5.2.3.1-5.2.3.4). To get a reasonable description of the virtual orbitals and to obtain a reasonable number of them into which to promote electrons, we need a basis set that is not too small. The use of the STO-1G basis in the above example was purely illustrative the smallest basis set generally considered acceptable for correlated calculations is the 6-31G, and in fact this is perhaps the one most frequently used for MP2 calculations. The 6-311G basis set is also widely used for MP2 and MP4 calculations. Both bases... 

These are the Hartree-Fock MOs and are kept fixed in the Cl calculation. 

CIS Configuration interaction including only single excitations from the Hartree-Fock MO determinant. A simple method for the calculations of excited states. 

However, the vacant Hartree-Fock molecular orbital (MO) obtained as a by-product of the ground-state calculations are of little use for describing the excited states of a molecule. This is due to the fact that the vacant Hartree-Fock MOs correspond to the motion of an excited electron in the potential field of all N electrons rather than of N - 1 electrons, as must be the case (Slater, 1963). Hunt and Goddard (HG) (1963) have proposed modifying the Hartree-Fock operator in such a way that it would be possible to describe the motion of an excited electron in the potential VN 1 ... 

With the 6-311+ + G basis, the total number of MOs is 568. There are 84 occupied MOs. Table 2 presents the VDEs of this anion calculated with a variable number of virtual orbitals retained according to the QVOS scheme [30]. The difference between the current QVOS code and the one described previously [30] is that in the latter all MOs had to be included, whereas in the current variant any orbital window can be chosen at the second-order step. Columns 1 and 2 of Table 2 list VDEs calculated either (1) with no reduction of the virtual space or (2) by simply omitting core and high-energy virtual MOs. In the latter case, the original set of canonical, Hartree-Fock MOs is employed. 

A classic illustration of the quality of a calculation is provided considering the effect of the basis set of say, a Hartree-Fock (HF) calculation. (See Sect. 3 for further discussion of the HF method). For MO schemes like HF theory, the variational principle states that the lower the computed total energy, the better the result [1]. The energy can be lowered by increasing the basis set size and so large basis set calculations are often described as good quality. However, this need not imply that experiment is also well reproduced. If the basic assumptions of the HF approximation are inappropriate, then it may not be possible to predict experimental data reliably, irrespective of the basis set size. As will be seen later, this is often the case for HF calculations on TM systems. 

To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

Just as several STOs are needed to give an accurate representation of Hartree-Fock AOs (Section 11.1), one needs more than one STO of a given n and / in the linear combination of STOs that is to accurately represent the Hartree-Fock MO. The primed and double-primed AOs in the extended-basis-set function are STOs with different orbital exponents. The 3dquantum number m = 0, that is, the 3do and 4/o AOs. The total energies found are -197.877 and -198.768 hartrees for the minimal and extended calculations, respectively. The experimental energy of F2 at Rg is -199.670 hartrees, so the error for the minimal calculation is twice that of the extended calculation. The extended-basis-set calculation is believed to give a wave function quite close to the true Hartree-Fock wave function. Therefore, the correlation energy in F2 is about -0.90 hartrees = -24.5 eV. 

FIGURE 13.20 Hartree-Fock MO electron-density contours for the ground electronic state of Li2 as calculated by Wahl. [A. CWahl,.Sctcwcc, 151,961 (l%6y, Scientific American, April l9J0,p.54 Atomic and Molecular Structure-. 4 Wall Charts, McGraw-Hill, 1970.]... 

Semiempirical molecular quantum-mechanical methods use a simpler Hamiltonian than the correct molecular Hamiltonian and use parameters whose values are adjusted to fit experimental data or the results of ab initio calculations an example is the Hiickel MO treatment of conjugated hydrocarbons (Section 16.3), which uses a one-electron Hamiltonian and takes the bond integrals as adjustable parameters rather than quantities to be calculated theoretically. In contrast, an ab initio (or first principles) calculation uses the correct Hamiltonian and does not use experimental data other than the values of the fundamental physical constants. A Hartree-Fock SCF calculation seeks the antisymmetrized product d> of one-electron functions that minimizes / dr, where H is the true Hamiltonian, and is thus an ab initio calcula-... 

A second example, H2 O, is depicted in Figure 3. Ib-e. But, before we can proceed with the discussion, we describe another useful orbital transformation localization of symmetry orbitals. Figure 3.1b shows the two bonding molecular orbitals (MOs) of H2O taken from a Hartree-Fock (HF) calculation. The 3aj orbital has even symmetry (++), while the Ibj orbital has odd symmetry (-F-). If we take the two linear combinations yT/2(3aj) -y TT flbj) of these orbitals, we see that two equivalent orbitals are produced (shown on the right side of the row). These are designated as and a j. because they are bond orbitals localized between O and the left and right H atoms, respectively. It is evident by inspection that each of these localized MOs closely resembles the a bond MO of OH shown in Figure 3.1a. 

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