Atomic orbital basis function

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

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... 

Atomic orbital basis functions have several indices, each referring to a different listing of these basis functions. In order to facilitate the correct index assignment in each case, several auxiliary quantities are defined. 

The formal vector cp (K) denotes the set of atomic orbital basis functions with centers at the original nuclear locations of the macromolecular nuclear configuration K, where the components cp(r, K) of vector q(K) are the individual AO basis functions. The macromolecular overlap matrix corresponding to this set cp (K) of AO s is denoted by S(K). The new macromolecular basis set obtained by moving the appropriate local basis functions to be centered at the new nuclear locations is denoted by cfcK ), where the notation cp(r, K ) is used for the individual components of this new basis set overlap matrix is denoted by S(K ). 

Contemporary basis sets are usually formed from atomic-orbital basis functions T), of contracted Gaussian form,... 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

All of the measurements employed the technique described above that involves the analysis of the isotope composition of 02 released from the carrier complexes in preequilibrated solutions. In addition, an established DFT method (mPWPW91)34 with the atomic orbital basis functions, Co, Fe, and Cl (the compact relativistic effective core potential basis CEP-31G),35 N and O (6-311G ), P (6-311G ), C(6-31G), and H (STO-3G),36 were used to calculate the 180 EIE in terms of actual and model structures. The latter approach has also been employed for hypothetical intermediates in enzymes as described below. 

Hartree-Fock-Roothaan SCF theory, using molecular orbitals expanded as linear combinations of atomic orbital basis functions (LCAO-MO theory)... 

Arrhenius activation energy 55 astronomical unit 110 asymmetry parameter 23 atmosphere 112 atomic mass 20, 41, 94 atomic mass constant 20, 41, 89 atomic mass unit 20, 41, 75, 89, 111 atomic masses of nuclides 98-104 atomic number 20, 44 atomic orbital basis function 17, 19 atomic scattering factor 36 atomic states 28 atomic units 76, 120 atomic weight 41, 94 atomization 51, 53 attenuance 32 atto 74... 

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

The orbitals (/> are assumed to be real and orthonormal, and are expressed as a linear combination of atomic orbitals (basis functions)... 

LCAO linear combination of atomic orbital basis functions... 

The key is that a single-center expansion of the transition density, implicit in a multipolar expansion of the Coulombic interaction potential, cannot capture the complicated spatial patterns of phased electron density that arise because molecules have shape. The reason is obvious if one considers that, according to the LCAO method, the basis set for calculating molecular wavefunctions is the set of atomic orbital basis functions localized at atomic centers a set of basis functions localized at one point in a molecule is unsatisfactory. 

However, the HF equations are still too difficult to solve in general, and so most commonly the molecular orbitals v/ (the spatial parts of the spin orbitals) are expanded as a linear combination of a finite set of atomic orbital basis functions < ), i.e. 

When g l,2) = we often express these integrals in short-hand notation as . It should be noted that the order of the creation and annihilation operators appearing in Eq. (1.16) must be as presented in order to guarantee that the proper sign will result when expectation and transition value matrix elements of such operators are formed. These spin orbitals are, in most practical applications, obtained as linear combinations of atomic orbital basis functions... 

Tn the previous section two II2 units were linked togctlicr in a planar geometry. If they arc joined together in a perpendicular manner (as in 5.8) tlien a tetrahedral II4 system results. Using the atomic orbital basis functions shown in 5.8, the orbital... 

Note that such calculations assume the total wave function as a single Slater determinant, while the resultant molecular orbital is described as a linear combination of the atomic orbital basis functions (MO-LCAO). Multiple Slater determinants in MO description project the configurational and post-HF methods and will not be discussed here. 

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