Atomic orbitals computations

MO Coefficients for Fluorine Atomic Orbitals Computed from F19 Hyperfine Interactions"... 

Election nuclear dynamics theory is a direct nonadiababc dynamics approach to molecular processes and uses an electi onic basis of atomic orbitals attached to dynamical centers, whose positions and momenta are dynamical variables. Although computationally intensive, this approach is general and has a systematic hierarchy of approximations when applied in an ab initio fashion. It can also be applied with semiempirical treatment of electronic degrees of freedom [4]. It is important to recognize that the reactants in this approach are not forced to follow a certain reaction path but for a given set of initial conditions the entire system evolves in time in a completely dynamical manner dictated by the inteiparbcle interactions. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

The technique for this calcu latioii in volves two steps. Th e first step computes the Hamiltonian or energy matrix. The elem en ts of this matrix are integrals involving the atomic orbitals and terms obtained from the Schrddiiiger equation. The m ost importan t con -... 

MP2 correlation energy calculations may increase the computational lime because a tw o-electron integral Iran sfonnalion from atomic orbitals (.40 s) to molecular orbitals (MO s) is ret]uired. HyperClicrn rnayalso need additional main memory arul/orcxtra disk space to store the two-eleetron integrals of the MO s. 

So called Ilydrogenic atomic orbitals (exact solutions for the hydrogen atom) h ave radial nodes (values of th e distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are n ot sensitive to these nodes and most simple calculation s use Slater atom ic orbitals ofthe form... 

The orbitals used for methane, for example, are four Is Slater orbitals of hydrogen and one 2s and three 2p Slater orbitals of carbon, leading to an 8 x 8 secular matrix. Slater orbitals are systematic approximations to atomic orbitals that are widely used in computer applications. We will investigate Slater orbitals in more detail in later chapters. 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

The quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. 

All molecular orbitals are combinations of the same set of atomic orbitals they differ only by their LCAO expansion coefficients. HyperChem computes these coefficients, Cp , and the molecular orbital energies by requiring that the ground-state electronic energy be at a minimum. That is, any change in the computed coefficients can only increase the energy. 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

To compute molecular orbitals, you must give them mathematical form. The usual approach is to expand them as a linear combination of known functions, such as the atomic orbitals of the constituent atoms of the molecular system. If the atomic orbitals, (Is, 2s, 2px, 2py, 2pz, etc.) are denoted as then this equation describes the molecular orbitals as linear combination of atomic orbitals (MO-LCAO) ... 

In an ab initio method, all the integrals over atomic orbital basis functions are computed and the Fock matrix of the SCF computation is formed (equation (61) on page 225) from the integrals. The Fock matrix divides into two parts the one-electron Hamiltonian matrix, H, and the two-electron matrix, G, with the matrix elements... 

Computation of the overlap integrals in the atomic orbital basis set. 

The self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

The CPHF equations are linear and can be determined by standard matrix operations. The size of the U matrix is the number of occupied orbitals times the number of virtual orbitals, which in general is quite large, and the CPHF equations are normally solved by iterative methods. Furthermore, as illustrated above, the CPHF equations may be formulated either in an atomic orbital or molecular orbital basis. Although the latter has computational advantages in certain cases, the former is more suitable for use in connection with direct methods (where the atomic integrals are calculated as required), as discussed in Section 3.8.5. 

The energy spectrum of the dodecaborides (YB,2, YbB,2, LuBj2> computed by the MO-linear combination of atomic orbitals (LCAO) method shows that the direction of the electron transfer is uncertain". ... 

The complex contains 72 atoms with 244 valence electrons distributed in 226 valence atomic orbitals. In order to reduce the computational effort, and to assess the contribution of the ligand 7r-orbitals to the overall spectrum, we examined a "reduced" model, see Figure 2, in which the benzene rings of the ligands are replaced by -HC=CH- groups. This model compound consists of... 

Consistently, the agreement with experiment is about 1% for geometric parameters and 5-10% for force constants. On the other hand, binding energies are found to be usually too large by l-2eV per atom. The computational efforts in the LDF approach scale with the third power in the number of orbitals. Consequently, the speed and memory capabilities of supercomputers will allow the study of systems with a hundred atoms and more. 

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