最佳定域分子轨道的新算法

本文提出了最佳群对称定域分子轨道（ＯＳＬＭＯＳ）的概念，报道了ＯＳＬＭＯｓ的生成方法。ＯＳＬＭＯｓ满足分子对称、等价与正交；同时最佳逅近经典的非正交价键型轨道，并可作为ＭＣＳＣＦ与ＣＩ方法中的单电子轨道。     

大体系多电子相关研究中应用群对称定域轨道的构想

大体系多电子相关研究中应用群对称定域轨道的构想周泰锦，刘爱民（厦门大学化学系，厦门３６１００５）关键词：组态相关，多构型自治叠代，多中心积分，群对称定域轨道，对称约化有关原子簇化合物及化学吸附、过渡态、激发态、催化反应等大体系的量子化学研究，对于探讨...     

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.     

镧系元素4f轨道在成键中的作用的理论研究

用密度泛函方法在冻结或不冻结4f轨道的条件下计算一系列含镧系元素双原子 分子,对结果进行分析,得出以下结论：镧系元素4f轨道按传统的化学键理论观点 是不直接参与成键的,但对成键有一定作用：通过与匹配物的轨道混合使键长变短 ,键能增加,一般可达百分之几。随着镧系原子序数的增加4f轨道对成键的贡献减 少。电负性高或价态高的匹配物对4f轨道的作用较强,4f轨道对成键的影响比较大 。对于重镧系元素,匹配物不是F或O时,4f轨道对成键的贡献相当小,可以看成芯 轨道,但对于轻稀土,在比较精确的计算中则应作为价轨道处理。镧系元素与氟结 合时,只有对靠近Yb的重镧系元素才可以把4f当作芯轨道,而与氧结合时即使对于 YbO把Yb4f作为芯轨道仍会带来较大误差。     

Local Hartree–Fock orbitals using a three‐level optimization strategy for the energy

Using the three‐level energy optimization procedure combined with a refined version of the least‐change strategy for the orbitals—where an explicit localization is performed at the valence basis level—it is shown how to more efficiently determine a set of local Hartree–Fock orbitals. Further, a core–valence separation of the least‐change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree–Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality. © 2013 Wiley Periodicals, Inc.     

Electron densities from the Brueckner Doubles method

Summary The concept of the Brueckner orbital is examined, following a resurgence of interest in wavefunctions constructed from them. The distinction between Self Consistent Field, Natural and Brueckner orbitals are discussed. Total electron densities are calculated for several examples, and correlation densities are studied. It is found that the Brueckner orbitals are more localised than SCF orbitals. The total electron density constructed from the Brueckner reference determinant with Brueckner orbitals gives qualitatively similar pictures as other correlated methods. Brueckner orbitals are found to show dissociation well.     

猪胰岛素的活力位置的电子结构分析

本文采用负因子计数方法对猪胰岛素分子作了完整分子的量子化学计算,其中对矩阵元的计算采用EHMO方法。我们将前线轨道的概念推广为靠近Fermi能级的能量区域中的分子轨道,将局域于少数氨基酸残基上的前线轨道简称为活性轨道。计算结果表明,猪胰岛素70％的活力敏感位置上局域有活性轨道。对全部结果的分析说明猪胰岛素的活性轨道和生物活力之间存在某些内在联系,但不是一一对应关系;猪胰岛素的三级构象的变化可以引起电子结构的变化;相同种类的氨基酸残基在分子的不同位置上可以有不同的活性轨道能级和轨道分布。     

Are there atomic orbitals in a molecule?

Effective atomic orbitals (AOs) have been calculated by the method of the "fuzzy atoms" analysis by using the numerical molecular orbitals (MOs) obtained from plane-wave DFT calculation, i.e., without introducing any atom-centered functions. The results show that in the case of nonhypervalent atoms there are as many effective AOs with non-negligible occupation numbers, as many orbitals are in the classical minimal basis set of the given atom. This means that, for nonhypervalent systems, it is possible to present the MOs as sums of effective atomic orbitals that resemble very much the atomic minimal basis orbitals of the individual atoms (or their hybrids). For hypervalent atoms some additional orbitals basically of d-type are also of some importance; they are necessary to describe the back-donation to these positive atoms. It appears that the d-type orbitals play a similar role also for strongly positive carbon atoms. The method employed here is also useful to decide whether the use of polarization functions of a given type is a matter of conceptual importance or has only a numerical effect.     

Hulthén and slater type 2d functions in some excited configurations of sulphur and phosphorus

It is shown that a substantial energy improvement is gained by the variational use of Hulthén orbitals, instead of single Slater orbitals, in the 3d shells of some excited configurations of sulphur and phosphorus. The energies obtained are close to those attained with two-term Slater functions. In some cases the radial distribution functions from Hulthén orbitals are as good an approximation of SCF radial distributions as those from two-term Slater orbitals. Single term 2d functions with only one parameter are found to give almost identical energies and radial distribution functions as those obtained from two-parameter Hulthén orbitals. It is shown that the relationship between one-term 2d orbitals and Hulthén orbitals gives a method of enforcing nuclear cusp conditions on the former with little effect on the energy.     

Dyson orbitals for ionization from the ground and electronically excited states within equation-of-motion coupled-cluster formalism: theory, implementation, and examples

Implementation of Dyson orbitals for coupled-cluster and equation-of-motion coupled-cluster wave functions with single and double substitutions is described and demonstrated by examples. Both ionizations from the ground and electronically excited states are considered. Dyson orbitals are necessary for calculating electronic factors of angular distributions of photoelectrons, Compton profiles, electron momentum spectra, etc, and can be interpreted as states of the leaving electron. Formally, Dyson orbitals represent the overlap between an initial N-electron wave function and the N-1 electron wave function of the corresponding ionized system. For the ground state ionization, Dyson orbitals are often similar to the corresponding Hartree-Fock molecular orbitals (MOs); however, for ionization from electronically excited states Dyson orbitals include contributions from several MOs and their shapes are more complex. The theory is applied to calculating the Dyson orbitals for ionization of formaldehyde from the ground and electronically excited states. Partial-wave analysis is employed to compute the probabilities to find the ejected electron in different angular momentum states using the freestanding and Coulomb wave representations of the ionized electron. Rydberg states are shown to yield higher angular momentum electrons, as compared to valence states of the same symmetry. Likewise, faster photoelectrons are most likely to have higher angular momentum.     

Are atoms sic to molecular electronic wavefunctions? II. Analysis of fors orbitals

Electronic wavefunctions that describe molecules in the full optimized reaction space (FORS) are multiconfigurational wavefunctions which are invariant under non-singular linear transformations of the occupied molecular orbitals. They offer therefore a considerably wider scope for orbital interpretations than the single-configuration Hartree-Fock approximation. For example they can be analyzed in terms of natural MOs and in terms of localized MOs. The latter turn out to be remarkably atomic in character and a new localization procedure can be formulated which yields atom-adapted molecular orbitals. These have the character of minimal-basis-set AOs that are optimally adapted to the molecular environment and furnish an unambigious atomic population analysis. On the other hand, chemically adapted molecular orbitals can be defined by an appropriate compromise between natural orbitals and localized orbitals. The freedom to use, as configuration-generating molecular orbitals, atom-adapted FORS MOs as well as chemically adapted FORS MOs makes FORS wavefunctions particularly suitable for chemical interpretations. The ensuing analysis establishes the minimal basis set (in molecule-adapted form) as a theoretically sound concept for the understanding of accurate molecular wavefunctions. An illustrative example is discussed.     

Variationally optimized basis orbitals for biological molecules

Numerical atomic basis orbitals are variationally optimized for biological molecules such as proteins, polysaccharides, and deoxyribonucleic acid within a density functional theory. Based on a statistical treatment of results of a fully variational optimization of basis orbitals (full optimized basis orbitals) for 43 biological model molecules, simple sets of preoptimized basis orbitals classified under the local chemical environment (simple preoptimized basis orbitals) are constructed for hydrogen, carbon, nitrogen, oxygen, phosphorous, and sulfur atoms, each of which contains double valence plus polarization basis function. For a wide variety of molecules we show that the simple preoptimized orbitals provide well convergent energy and physical quantities comparable to those calculated by the full optimized orbitals, which demonstrates that the simple preoptimized orbitals possess substantial transferability for biological molecules.     

Pipek–Mezey localization of occupied and virtual orbitals

Recent advances in orbital localization algorithms are used to minimize the Pipek–Mezey localization function for both occupied and virtual Hartree–Fock orbitals. Virtual Pipek–Mezey orbitals for large molecular systems have previously not been considered in the literature. For this work, the Pipek–Mezey (PM) localization function is implemented for both the Mulliken and a Löwdin population analysis. The results show that the standard PM localization function (using either Mulliken or Löwdin population analyses) may yield local occupied orbitals, although for some systems the occupied orbitals are only semilocal as compared to state‐of‐the‐art localized occupied orbitals. For the virtual orbitals, a Löwdin population analysis shows improvement in locality compared to a Mulliken population analysis, but for both Mulliken and Löwdin population analyses, the virtual orbitals are seen to be considerably less local compared to state‐of‐the‐art localized orbitals. © 2013 Wiley Periodicals, Inc.     

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.     

Optimization of molecular electronic structure calculations. 1. Local symmetry and local symmetricized orbitals

A formalism is suggested of the so-called local symmetricized orbitals to be used for the construction of a symmetricized basis in molecular electronic structure calculations. The local symmetricized orbitals are defined as additive contributions to the symmetry orbitals of a molecule that arise from symmetry operations of a corresponding atom. The local symmetricized orbitals are transformed according to the irreducible representations of the molecular symmetry group. This approach appears to be most suitable for the optimization of quantum mechanical calculations accounting for the spatial symmetry of compounds under consideration. This fact is due to the formalism of the local symmetricized orbitals that explicitly accounts for the local symmetry of basis function centers, which is essential for such optimization.     

An extension of the pairing theorem

An extended pairing scheme is presented which ensures the fulfillment of pairing conditions not only between the sets of occupied orbitals for spin α and for spin β, but also between their orthogonal complements, i.e., the sets of virtual orbitals for spin α and spin β, as well as between occupied orbitals for spin α and virtual orbitals for spin β and between virtual orbitals for spin α and occupied orbitals for spin β. It is shown that the extended pairing properties are suggested by some aspects of the construction of alternant molecular orbitals. The algorithm for singular value decomposition of rectangular matrices is proposed for use in practical implementations of the (extended) pairing scheme.     

价键方法中的非活性轨道优化新算法

本文通过引进一组正交的辅助非活性轨道和与它正交的辅助活性轨道,将价键理论方法中的冻核近似推广到轨道非正交的情形,得到了体系能量及其对非活性轨道的梯度解析表达式,简化了价键自洽场方法中非正交轨道能量梯度的计算．该方法的标度为（Na＋1）m^4,其中Na和m分别是活性轨道和基函数的个数．分析表明,与现有的其他算法相比较,该方法具有更低的计算标度,因而计算效率更高．     

Orbital signatures of methyl in L-alanine

Molecular orbital signatures of the methyl substituent in L-alanine have been identified with respect to those of glycine from information obtained in coordinate and momentum space, using dual space analysis. Electronic structural information in coordinate space is obtained using ab initio (MP2/TZVP) and density functional theory (B3LYP/TZVP) methods, from which the Dyson orbitals are simulated based on the plane wave impulse approximation into momentum space. In comparison to glycine, relaxation in geometry and valence orbitals in L-alanine is found as a result of the attachment of the methyl group. Five orbitals rather than four orbitals are identified as methyl signatures. That is, orbital 6a in the core shell, orbitals 11a and 12a in the inner valence shell, and orbitals 19a and 20a in the outer valence shell. In the inner valence shell, the attachment of methyl to glycine causes a splitting of its orbital 10a' into orbitals 11a and 12a of L-alanine, whereas in the outer valence shell the methyl group results in an insertion of an additional orbital pair of 19a and 20a. The frontier molecular orbitals, 24a and 23a, are found without any significant role in the methylation of glycine.     

A theory of molecules in molecules IV. Application to the Hydrogen Bonding Interaction in NH3 · H2O

The theory of molecules in molecules introduced in previous articles is applied to study the hydrogen bonding interaction between an ammonia molecule as proton acceptor and a water molecule as proton donor. The localized orbitals which are assumed to be least affected by the formation of the hydrogen bond are transferred unaltered from calculations on the fragments NH₃ and H₂O, the remaining orbitals are recalculated. A projection operator is used to obtain orthogonality to the transferred orbitals. Additional approximations have been introduced in order to be able to save computational time. These approximations can be justified and are seen to lead to binding energies and bond lengths which are in satisfactory agreement with the SCF values. The point charge approximation for the calculation of the interaction energy between the two sets of transferred localized orbitals is, however, not applicable in this case. An energy analysis of the effect of the hydrogen bond on the localized orbitals of the two fragments is given.     

An efficient method for calculating maxima of homogeneous functions of orthogonal matrices: applications to localized occupied orbitals

We present here three new algorithms (one purely iterative and two DIIS-like [Direct Inversion in the Iteractive Subspace]) to compute maxima of homogeneous functions of orthogonal matrices. These algorithms revolve around the mathematical lemma that, given an invertible matrix A, the function f(U)=Tr(AU) has exactly one local (and global) maximum for U special orthogonal [i.e., UU(T)=1 and det(U)=1]. This is proved in the Appendix. One application of these algorithms is the computation of localized orbitals, including, for example, Boys and Edmiston-Ruedenberg (ER) orbitals. The Boys orbitals are defined as the set of orthonormal orbitals which, for a given vector space of orbitals, maximize the sum of the distances between orbital centers. The ER orbitals maximize total self-interaction energy. The algorithm presented here computes Boys orbitals roughly as fast as the traditional method (Jacobi sweeps), while, for large systems, it finds ER orbitals potentially much more quickly than traditional Jacobi sweeps. In fact, the required time for convergence of our algorithm scales quadratically in the region of a few hundred basis functions (though cubicly asymptotically), while Jacobi sweeps for the ER orbitals traditionally scale as the number of occupied orbitals to the fifth power. As an example of the utility of the method, we provide below the ER orbitals of nitrated and nitrosated benzene, and we discuss the chemical implications.