On the choice and definition of atomic-orbital bases. I. General considerations; promotion and hybridization; electric dipole moments

After a brief discussion of the physical significance of the choice of the basis in molecular calculations, the nature and definition of an atomic-orbital basis for use in limited calculations is discussed, in view of the possibility of replacing, say, ordinary 2s and 2p Slater orbitals by appropriate hybridized-promoted atomic orbitals. It is indicated that, if the orbitals must be defined in connection with a given interpretation scheme for the behavior of molecules, hybridization and promotion may be necessary. The two kinds of conditions one may wish to impose on a restricted atomic-orbital set are explicitly considered. The first is that the atomic orbitals should be hybrids directed along the bonds and at the same time satisfy the maximum overlap criterion; the other is the requirement that the atomic orbitals should be such that the electric dipole moment of a polyatomic molecule described in terms of a semiempirical bond-orbital scheme should be expressed as the dipole moment of the system of bond charges located at the nuclei. The latter condition is treated in detail, showing that it implies a cancellation of atomic and overlap moments. The equations defining the atomic orbitals satisfying the condition in question are given. In the course of the mathematical treatment some general results concerning the expression of the dipole moment of a molecule and the definition of net atomic charges are given, showing that, for systems where overlap integrals are low, the atomic populations can be taken as sums of the squares of the coefficients of orthogonalized atomic orbitals. Applications of the results will be presented in part II.     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

An intrinsic localization procedure for active CAS SCF orbitals

An intrinsic localization criterion for the active (valence) orbitals of a CAS SCF wavefunction is presented. The localization criterion is based on minimization of the energy of a “perfect pairing” configuration. Equations for carrying out the localization in terms of an exponential transformation are developed. The technique can easily be incorporated into any MC SCF program. The CAS SCF wavefunction obtained using these localized active orbitals corresponds to a full VB calculation where the VB structures are built from orthogonal “molecule adapted minimal basis set atomic orbitals” and thus offers an interpretational advantage over the use of canonical CAS SCF orbitals. The method is applied to the 1,3-dipole, nitrone.     

Charge distributions and multipole moments in molecules

The consideration of multipole moments is suggested as a new criterion for the validity of assignments of atomic charges in molecules. The total quadrupole and octupole moments generated by our definition of atomic charges are compared with the exact moments of the underlying wavefunction for various basis sets in selected diatomics. The analysis includes also total overlap and total dipole moment partitioning as well as 1σ MO overlap partitioning. All considerations together allow us to assess the validity of our charge definition as compared to Mulliken's and Löwdin's and the quality of the basis set.     

Maximum overlap symmetry orbitals

Based on the principle of maximum overlap, a new simple method is suggested for constructing the symmetry orbitals of arbitrary molecules and the delocalized molecular orbitals of molecules that do not involve the rings with an odd number of atoms. All these orbitals, called “maximum overlap symmetry orbitals,” are determined by an extended maximum overlap criterion and form the bases for the irreducible representations of the molecular point symmetry group. The theoretical analysis and the numerical results show that the obtained molecular orbitals are close to those obtained from the customary LCAO method, and calculation by the proposed method requires less computing time than does the LCAO method, thus illustrating a fact that the method is not only a reasonable approximation of the LCAO method, but simpler and feasible in large molecular systems.     

等性扩展基杂化轨道的构造

杂化轨道理论近来有了长足的发展,杂化轨道的构造方法主要有群论方法最大重叠方法,自然杂化轨道法和其它以分子轨道为基础构造杂化轨道的方法等。其中最大重叠杂化轨道不仅满足正交化条件而且能较定量地考虑到配体轨道的作用,因而已经得到广泛应用。本文在最大重叠原理的基础上得到了扩展基杂化轨道的解析形式。扩展基杂化轨道对一给定几何构型的分子M(X_1X_2…X_n),其中心原子M的n个杂化轨道与诸配体{X_i}形成一组方向键。M的杂化轨道(HO)和原子轨道(AO)可分别作为该分子对称操作群的表示之基。这两种不同基的表示进行约化之后,属于同一不可约表示的HO和AO是线性相     

键能的分子轨道理论研究2: 键能与Mulliken布居对键强度的 判断的比较

用键能E~A~B和Mulliken布居对化学键强度的判别进行了分析比较。结果表明,键能判据比Mulliken布居判据所得结论更符合实际情况。作为衡量原子间化学键强度的尺度,不仅应考虑原子轨道间的布居因素,还应考虑分子轨道(或原子轨道)的能量因素。     

The Fock Matrix Analysis for Atomic Orbitals in Molecular Orbitals I. A New Look on the Covalent Bond in H2?

The character of the molecular orbitals can be better accounted for in terms of molecular adapted atomic orbitals and the Fock matrix expanded in these atomic orbital sets. A clean‐cut and unique criterion for the diradicals and the covalent bonds can be given for the molecular orbitals in both restricted and unrestricted Hartree‐Fock wavefunctions. Instead of the picture that overlap charge migrates into the bonding region, the new analysis displays another picture that the charge densities for the electrons with α and β spins give rise to two opposite spin density shifts. If the α one shifts from atom A toward atom B then it is vice versa for the β one. The spin density shifts proceed until the bonding molecular orbitals form.     

Localized orbitals based on the fermi hole

The Fermi hole provides a direct (non-iterative) method for tansforming canonical SCF molecular orbitals into localized orbitals. Except for simple overlap integrals required to maintain orthogonality, this method requires no integrals over orbitals or basis functions. This method is demonstrated by application to a furanone (C₄H₄O₂), methylacetylene, and boron trifluoride. The results of these calculations are compared to those determined by the orbital centroid criterion of localization.     

An orbital localization criterion based on the theory of "fuzzy" atoms

This work proposes a new procedure for localizing molecular and natural orbitals. The localization criterion presented here is based on the partitioning of the overlap matrix into atomic contributions within the theory of "fuzzy" atoms. Our approach has several advantages over other schemes: it is computationally inexpensive, preserves the sigma/pi-separability in planar systems and provides a straightforward interpretation of the resulting orbitals in terms of their localization indices and atomic occupancies. The corresponding algorithm has been implemented and its efficiency tested on selected molecular systems.     

The generalized method for construction of hybrid orbitals by the maximum overlap method

The calculation of the best hybrid atomic orbitals by the maximum overlap method is elaborated for the molecular system of polycentric character. The variational procedure is used to maximize the weighted sum of the overlap integrals.     

The Fock Matrix Analysis for Atomic Orbitals in Molecular Orbitals II. The Electronic Structure of N2 Molecules

Weinhold's natural hybrid orbitals can be chosen as the molecular adapted atomic orbitals to build the canonical molecular orbitals of N₂ molecules. The molecular Fock matrix expanded in the natural hybrid orbitals can reveal deeper insight of the electronic structure and reaction of the N₂ molecule. For example, the magnitude of F_ab can signify the bonding character of the paired electrons as well as the diradical character of the unpaired electrons for both σ‐ and π‐types. Discarding the concept of the overlap between non‐orthogonal atomic orbitals, the different orbitals for different spins in the unrestricted Hartree‐Fock wavefunction reveal that there are three pairs of opposite spin density flows between two atoms, which proceed until the bonding molecular orbitals form.     

簇电荷对金属—金属键影响的量子化学研究

在探讨过渡金属原子簇化合物的金属——金属键的本质时,簇电荷的影响已引起人们的注意。簇电荷对金属——金属键的作用比较复杂,其中有价电子的成键效应和金属原子氧化数变化所产生的电荷效应。键长与簇电荷之间很难找到简单的关系。Cotton等曾对此问题做过初步讨论,但尚缺定量或半定量的理论计算依据,本文采用改进的电荷自洽EHMO程序(MAD—SCCO-EHMO)计算一系列Mo,Tc,Ru,Rh,和Re等二核簇的电子结构,根据M(?)lliken重迭集居分析,讨论簇电荷对金属——金属键的影响。     

Determination of the charge distribution in nonmetallic crystals. I. Theory and application to tetrahedrally coordinated solids

A SCF method using localized molecular orbitals which are built up on hybrid atomic orbitals is proposed to obtain the charges in infinite crystals. Hybrid orbitals are built up on a minimal STO basis set. The formalism has been adapted in order to take into account the periodicity of the system and its infinite size by introducing the Madelung constant. The total energy is given by an infinite sum of terms each corresponding to the energy of a bond in the crystal field. Minimizing this bond energy with respect to eigenvectors it is straightforward to obtain the electronic charges, whence the polarity, i.e., the ionicity, of bonds. In this first paper, we study and discuss the polarity of bonds in zincblende and wurtzite-type compounds built up on first and second row elements. Our values are coherent between themselves and in agreement with other authors' results. The connection with electronegativity and polarizability is discussed.     

Interpretation of the indirectly induced field at nonmagnetic atoms in Fe-Y garnets

The indirectly induced magnetic fields at Sn⁴⁺ replacing Fe³⁺ in octahedral positions are derived in the MO LCAO [molecular orbitals by linear combinations of atomic orbitals] semiempirical approximation, with self-consistent charges on the atoms, for a part of the crystal containing Sn⁴⁺ and Fe³⁺ (in tetrahedral positions), the two being linked via oxygen ions. The absence of conduction is due to the very restricted overlap between the MO of the parts. The results explain the observed induced field and also predict certain possible effects, in particular unpaired spin density on other lattice ions.     

Valence-bond calculations on ZNO and HGO using integrals computed through the semiempirical AM1 method

Valence-bond calculations have been carried out on ZnO and HgO using a basis set of Slatertype atomic orbitals and the one- and two-electron integrals as computed in the semiempirical AM 1 molecular orbital method. The zero differential overlap approximation has been used to calculate integrals between atomic orbital Slater determinants using the rules for matrix elements between determinants formed by orthogonal orbitals. Diabatic and adiabatic curves have been analyzed for the two systems, and results compared with molecular orbital AM 1 results. © 1992 John Wiley & Sons, Inc.     

Orbital Interaction and the Mulliken-Walsh Diagram for AH2 Systems

We define θ- (or R-) bonding (or antibonding) character of the molecular orbitals according to the slopes of the orbital energy curves when the internuclear angle (or internuclear distance) is varied. So far the slope of the orbital curve has only been accounted for by the qualitative argument based on two factors: the orbital overlap and the s-p mixing. We employ the bond orbitals instead of the usual atomic orbitals as the basis set to analysis the character of the molecular orbitals. The Fock matrix in the bond orbital basis can then quantitatively account for the effects of both the overlap and s-p mixing factors. Our analysis also show that a third factor, the orbital interaction, is essential to account for both the “typical” and “abnormal” behavior of the slopes.     

Multicenter point charge model for high-quality molecular electrostatic potentials from AM1 calculations

The natural atomic orbital/point charge (NAO-PC) model based upon the AM1 wave function has been developed to calculate molecular electrostatic potentials (MEPs). Up to nine point charges (including the core charge) are used to represent heavy atoms. The positions and magnitudes of the eight charges that represent the atomic electron cloud are calculated from the natural atomic orbitals (NAOs) and their occupations. Each hybrid NAO is represented by two point charges situated at the centroid of each lobe. The positions of the centroids and the magnitudes of the charges were obtained by numerical integration of the Slater-type hybrids and the results used to set up polynomials and look-up tables that replace the integration step in the actual MEP calculation. The MEPs calculated using this method are found to be in better agreement with those obtained using RHF/6-31G* than those obtained from the AM1 wave function using Coulson charges or with MOPAC-ESP. The MEP calculations are extremely fast and have, for instance, been incorporated into an interactive graphics package. © 1993 John Wiley & Sons, Inc.     

Structure and electronic properties of the surface of alkali metal peroxides

The CRYSTAL09 program with the implemented B3PW hybrid density functional in a localized basis of atomic orbitals is used to determine the atomic and electronic structure of the surface of lithium, sodium, and potassium peroxides. Geometric parameters, surface energies, partial densities of states, electron density distributions, overlap populations, and atomic charges are calculated. It is found that the geometry relaxation has a characteristic depth up to ～10 Å, while the surface states are located in the upper layers at a depth up to ～2.5 Å. Structural displacements of atoms do not exceed ～ 0.2 Å; the charge of the upper surface layers is positive, whereas the energy state shifts relative to the bulk ones can reach ～1 eV. The surface energy of peroxides decreases with an increase in the atomic number of the cation.