Natural spin orbital analysis of diatomic molecular wave functions in terms of generalized diatomic orbitals

The original idea of the model applied to HeH⁺ excited states is: One electron occupies a diatomic orbital similar to the HeH⁺⁺ ground state 1s function. The other electron occupies an orbital which can be represented by a linear combination of functions similar to H ₂ ⁺ excited state functions. One or two screening parameters are variationally optimized to compensate for the smallness of the one-electron basis.CI calculations have been performed for five excited HeH⁺ states covering a wide range of internuclear distances. The CI wave functions have been submitted to a natural spin orbital analysis. The strongly occupied NSOs are compared with the original model functions.Dedicated to Professor Hermann Hartmann on occasion of his 65th birthday on May 4th, 1979.     

A new operator formulation of the many-electron problem for molecules

An n-electron operatorX _n , called a wave operator, is associated with a 2n-electron molecular wave function. Electron densities and energy are written in terms ofX _n . An equation defining an exact wave operator is found. Thus, a 2n-electron vector problem (for the wave function) is rigorously reduced to an n-electron operator problem. Conditions are formulated which guarantee thatX _n corresponds to a state with a given spin. The configuration-interaction problem is considered and methods of approximate construction ofX _n are discussed. In particular, a matrix algorithm is proposed for calculations in the two-body approximation. A generalizaton of the approach to the case of systems with an odd number of electrons is given. The waveoperator model developed forms a general basis for construction of covariant electron models of molecules.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 25, No. 1, pp. 1–12, January–February, 1989.     

The influence of basis sets on wave function tails

Wave function tails are analyzed quantitatively by investigating the dependence of exterior electron density (EED ) on basis sets; the EED is defined as the integrated electron density outside the repulsive molecular surface. Ab initio MO calculations with large scale basis sets were performed to establish the benchmark order of EED values for valence orbitals of some simple molecules. It is found that very popular basis sets, such as 4-31G, which are determined by energy optimization, are inferior in describing the wave function tails to some similar size basis sets, such as MIDI -4, which are obtained by least-squares fit to near Hartree-Fock atomic functions. Further the EED values for atomic 2s functions are shown to be unfavorably smaller than those for atomic 2p functions when the same value is used for the exponent α in the GTO basis sets. This indicates that the frequently used constraint α_s = α_p is not appropriate for describing wave function tails with medium-size basis sets. Deficiencies in the energy-optimized basis sets are found to become more serious for molecules including heavier atoms.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

A general nonlinear expansion form for electronic wave functions

A new expansion form is presented for electronic wave functions. The wave function is a linear combination of product basis functions, and each product basis function in turn is formally equivalent to a linear combination of configuration state functions that comprise an underlying linear expansion space. The expansion coefficients that define the basis functions are nonlinear functions of a smaller number of variables. The expansion form is appropriate for both ground and excited states and to both closed and open shell molecules. The method is formulated in terms of spin-eigenfunctions using the graphical unitary group approach (GUGA), and consequently it does not suffer from spin contamination.     

Projected electron density method of π-eletron systems II. Excited states

A perturbation theory based on the time-dependent Schrödinger equation is presented; Coulombic interactions are taken into account and spin properties are neglected. Using wave functions given by the projected electron density method described in Part I as a basis set the energies of excited π-electron states are calculated. For a series of porphyrin compounds the electronic spectra are calculated and are found to be in good agreement with experiment.     

Optical oscillator strengths for lithium

A simple two-parameter analytic potential adjusted so as to reproduce the experimental energy levels is used to generate wave functions for the ground and excited states of the lithium atom. Using these wave functions in conjunction with the Born approximation and the Russell–Saunders LS-coupling scheme, we calculate the optical oscillator strengths for various excitations from the 1s²2s(²S_1/2) ground state. The results are compared to experiment and other calculations.     

Optical oscillator strengths for isoelectronic ions of nitrogen

An analytic atomic active electron model potential adjusted to experimental single-particle energy levels is used to generate wave functions for the valence and excited states of O¹⁺, F²⁺, Ne³⁺, Na⁴⁺, and Mg⁵⁺. Using these wave functions in conjunction with the Born approximation and the LS-coupling scheme, we calculate optical oscillator strengths for excitations from the 1s²2s²2p³(⁴S_3/2) ground state. The results are compared to experimental data and other calculations. Systematic trends along the isoelectronic sequence are discussed.     

Unitary symmetry and classification of the states of <Emphasis Type="Italic">n</Emphasis>-spin clusters. Magnetic and thermodynamic parameters

The unitary symmetry and classification of spin clusters by spin momenta S are considered on the basis of reduction of the full linear group to unitary groups U _{2s + 1} and orthogonal rotation group R ₃. Reduction of the permutation group P _n of n spins to the point group of the cluster is applied to the classification of the spatial states of a spin cluster with the use of permutation quantum numbers introduced in this work and the Young diagrams of the permutation group P _n . Examples of the classification of spin systems with spins s = 1/2, 1, 3/2, 2, and 5/2 with U _{2s + 1} × P _n groups (n = 5–15) are reported. This classification is common for all spin clusters and is the same for both cyclic clusters and 3D clusters with symmetry groups of a crystal. On the basis of this classification, the magnetic and thermodynamic parameters of a spin system are calculated as a function of the number of spins and temperature. For s = 1/2 clusters, the analytical formulas are derived for magnetic susceptibility, internal energy, heat capacity, and entropy as a function of quantum numbers for a cluster with any number of spins, and their dependences on temperature and the number of atoms are considered.     

Quasi-homopolar levels of hydrocarbons with conjugate bonds: Series for projected π-electron operators

These are states that go over to the 2p_z states of the neutral atoms as the latter recede to infinity; they include the ground state and most of the lower excited states. Then Schrödinger's equation and the operators for the physical quantities may be projected on the space of spin functions. A method is given for calculating the projected hamiltonian and operators as a rapidly convergent series in the number of interacting centers. Pair interactions are shown to play the main part in the spin hamiltonian. The convergence is examined for the series for the momentum and spin-density operators. Schrödinger's equation with the spin hamiltonian then gives a complete solution of the problem; as in the valence-bond method, the task is facilitated by the fact that the subspaces of defined system spin may be distinguished in spin space. A method is given for selecting the states from the measured terms for the molecule. It is shown that all absorption lines corresponding to excitation of such states should be weak for alternant hydrocarbons.     

Extremal electron pairs

Extremal pair functions for an n-electron wave function of a closed-shell state are defined as linear combinations of spin-orbital-product pair functions that make some functionals (e.g., r²₁₂ or r⁻¹₁₂) extremal. They are related to the natural spin geminals in the uncorrelated limit and are useful both for an analysis of wave functions in view of an understanding of the chemical bond and for the treatment of electron correlation. Numerical examples are shown and discussed for He₂ as well as the 10-electron systems Ne, HF, H₂O, NH₃, and CH₄. © 1996 John Wiley & Sons, Inc.     

Charge densities for singlet and triplet electron pairs

Analytic properties of charge densities associated with singlet and triplet electron pairs, ρ₀( r ) and ρ₁( r ), are presented. In an N‐electron system with total spin S, distributions ρ₀( r ) and ρ₁( r ) are independent of the spin projection quantum number M (spin rotation invariance), as opposed to the usual spin‐up and spin‐down electron densities, ρ_α( r ) and ρ_β( r ). We derive equations showing that in the case of a wave function which is a spin‐eigenfunction, ρ₀( r ) and ρ₁( r ) are linear combinations of the total charge density ρ( r ) and the uncompensated spin density ρ_s( r )=[ρ_α( r )−ρ_β( r )]/2M. For a wave function which is not an eigenfunction of $\mathcal{S}^{2}$, no such relationship exists. In a related discussion, a definition of the high‐spin solution corresponding to a given spin‐unrestricted Hartree–Fock wave function is proposed, and a notion of effectively unpaired electrons is introduced. The distributions ρ₀( r ) and ρ₁( r ) are shown not to be invariant under spin coupling between isolated systems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 651–660, 2000     

Theoretical approach to fluorine substitution in X2NO and X2CN free radicals. Comparison between ab initio UHF and RHF + perturbation treatments

Non-empirical calculations have been performed to analyze the effects of fluorine substitution on the geometry and electronic properties of two series of π and σ radicals. Both UHF and RHF + perturbation methods have been used and the results are compared as a function of the basis set quality. As concerns geometry and spin-free electronic properties the results are independent of the UHF or RHF formalism, but highly sensitive to the basis set. The STO-3G basis is unable to reproduce even general trends. Polarization functions always play a relevant role and correlation effects seem not negligible at least for fluorine-containing radicals. The molecular shape of π radicals changes from a planar to a pyramidal geometry when increasing the electronegativity of the substituents. On the contrary, σ radicals always remain planar. Unprojected UHF spin densities are closer to the RHF + perturbation results for small spin contamination (X₂NO). On the contrary, it is the projected UHF spin density which is in better agreement with the RHF + perturbation value for large spin contamination (X₂CN). No simple correlation can be found between spin densities and gross atomic spin populations. For H₂NO the spin density at nitrogen is smaller than at the oxygen nucleus, but substitution may enhance or reverse this trend.     

One-Electron Pseudo-Potential Determination of Stable Isomers of the Li*Ar n Excited Clusters: Absorption Spectra

We present pseudo-potential calculations of geometrical structures of stable isomers of LiAr _n clusters with both an electronic ground state and excited states of the lithium atom. The Li atom is perturbed by argon atoms in LiAr _n clusters. Its electronic structure obtained as the eigenfunctions of a single-electron operator describing the electron in the field of a Li⁺Ar _n core, the Li⁺ and Ar atoms are replaced by pseudo-potentials. These pseudo-potentials include core-polarization operators to account for the polarization and correlation of the inert core with the valence Lithium electron [J Chem Phys 116, 1839 1]. The geometry optimization of the ground and excited states of LiAr _n (n = 1–12) clusters is carried out via the Basin-Hopping method of Wales et al. [J Phys Chem 101, 5111 2; J Chem Phys 285, 1368 3]. The geometries of the ground and ionic states of LiAr _n clusters were used to determine the energy of the high excited states of the neutral LiAr _n clusters. The variation of the excited state energies of LiAr _n clusters as a function of the number of argon atoms shows an approximate Rydberg character, corresponding to the picture of an excited electron surrounding an ionic cluster core, is already reached for the 3s state. The result of optical transitions calculations shows that the absorption spectral features are sensitive to isomer structure. It is clearly the case for transitions close to the 2p levels of Li which are distorted by the cluster environment.     

Study of electron impact excitation of atoms through lasers

We discuss three different experiments for studying electron-excitation of atoms where lasers have been used in combination. These are stepwise electron–photon excitation, superelastic electron scattering from laser excited atoms and excitation of atoms using spin polarized electrons produced by lasers. We present distorted wave calculations and compare our results with the recently reported such experimental measurements. In particular, the results for the alignment and orientation of the excited n ²P states of K ($n=4$) and Rb ($n=5$) atoms and the spin parameters for the lowest excited ¹P₁ and ³P_0,1,2 states of argon by polarized electrons are presented and discussed.     

Young operator methods for fermion systems

Summary Alternative methods to the standard Young technique for the construction of Fermion wave functions in the spin orbital formalism are presented and shown to be equivalent to the standard technique. To develop these methods: (i) the starting or primitive function is factored into spin and spatial parts, (ii) the conjugacy feature required to satisfy the antisymmetry principle is exploited, (iii) the necessary commutation relations with the Fermion antisymmetrizer are shown to hold and (iv) the one-to-one correspondence between the independent picture of the Young tableaux and the independent Slater determinants is used. This last feature has the advantage of reducing all three methods to rapid efficient graphical procedures. Each method is analyzed to consider the amount of labor involved to carry it out. Several examples of the methods are given for constructing both electronic wave functions and spin functions.     

Doubly excited3Se,3De and3Ge states of two-electron atomic systems

Summary Time-dependent perturbation theory has been applied to calculate the doubly excited triplet statesNsns:³S^e,Npnp:³D^e andNdnd:³G^e (N=2, 3, 4,n=N+1, ... ,5) for He, Li⁺, Be²⁺ and B³⁺. A time-dependent harmonic perturbation causes simulataneous excitation of both the electrons with a change of spin state. The doubly excited energy levels have been identified as the poles of an appropriately constructed linearized variational functional with respect to the driving frequency. In addition to the transition energies, effective quantum numbers of these doubly excited states have been calculated and analytic representations of their wave functions are obtained. These are utilized to estimate the Coulomb repulsion term for these states which checks the consistency of the wave functions. These wave functions may also be used for calculating other physical properties of the systems.     

Spin-projected Extended Hartree-Fock using a Valence Bond approach

Summary We describe spin-projected Extended Hartree-Fock calculations, performed with a Valence Bond Self-Consistent Field program. Potential energy curves are given for BH, BeH, and N₂. For BH the EHF function ranks well with the corresponding Spin-coupled and full CI wave functions. For BeH, the EHF function introduces spin contamination in the separated Be atom due to the rigidity of the wave function. This results in an inferior potential energy curve compared to Spin-coupled and full CI. The triple bond breaking in N₂ is again nicely described by EHF. The Extended Hartree-Fock method as suggested by Löwdin can be a feasible tool in describing bond breaking.     

Spin projections for even electron systems

A new method is presented for obtaining the spin eigenfunctions of 2n electron systems in the spin state S and M_S = 0. Using a modified Young operator the function for the state S, M_S ( = S) of the system is first projected out. The projected function is then symmetrized over the last n particles and the weight lowering operator ? is then applied to it, resulting in a projection of the state S, 0. To within a multiplicative factor, the resulting function is identical with the one resulting from the vector coupling methods.     

Conformational dependence of the 1Lb and n → π* rotatory strengths in α-substituted phenylacetic acids

The chiroptical properties of a series of chiral phenylacetic acid derivatives are examined on a theoretical model in which electronic rotatory strengths are calculated directly from molecular wave functions derived from semiempirical molecular orbital calculations. The CNDO/S SCF-MO model is used to calculate ground state wave functions and excited states are constructed in the virtual orbital-configuration interaction approximation. Of special interest are the rotatory strengths associated with the ¹L_b transition of the phenyl chromophore and the n → π^* transition of the carboxyl chromophore. Calculations are carried out on a large number of conformational isomers of the compounds: α-methyl phenylacetie acid (and its methyl ester), α-methylmandelic acid (and its methyl ester), and mandelic acid (and its methyl ester). The dependence of rotatory strength (¹L_b and n → π^*) on conformational variables is examined and discussed, and comparisons between available experimental data and the calculated results are made. Coupling between the phenyl and carboxyl chromophoric moieties is considered and possible spectra-structure relationships are examined.