Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Radial and angular correlation in heliumlike ions

In the framework of nonrelativistic variational formalism a new type of basis set is proposed, to estimate separately the effect of radial and angular correlations on the ground‐state energy for helium isoelectronic sequence H^? to Ar¹⁶⁺. Effect of radial correlation is incorporated by using multiexponential functions arising from product basis sets suitably formed out of Slater‐type one‐particle orbitals. The angular correlation can be switched on by incorporating an expansion in terms of basis involving interparticle coordinates. With a set of six‐term Slater‐type one‐particle basis and five‐term interparticle expansion, the ground‐state energy of helium is estimated as ?2.9037236 (a.u.) compared with the multiterm variational estimates ?2.9037244 (a.u.) due to Pekeris and Thakkar and Smith and Drake. Matrix elements of different operators in the ground state have been calculated and found to be in good agreement with available accurate results. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Ab initio quasirelativistic calculations on electronic transitions in ICl by the multireference many‐body perturbation theory

A quasirelativistic perturbative method of ab initio calculations on ground and excited molecular electronic states and transition properties within the relativistic effective core potential approximation is presented and discussed. The method is based on the construction of a state‐selective many‐electron effective Hamiltonian in the model space spanned by an appropriate set of Slater determinants by means of the second‐order many‐body multireference perturbation theory. The neglect of effective spin–orbit interactions outside of the model space allows the exploitation of relatively high nonrelativistic symmetry during the evaluation of perturbative corrections and therefore dramatic reduction of the cost of computations without any contraction of the model‐space functions. One‐electron transition properties are evaluated via the perturbative construction of spin‐free transition density matrices. Illustrative calculations on the X0⁺ ? A1, B0⁺, and (ii)1 transitions in the ICl molecule are reported. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Configuration interaction calculations on the 2P ground state of boron atom and C+ using Slater orbitals

Configuration Interaction (CI) calculations on the ground ²P state of boron atom are presented using a wave function expansion constructed with L‐S eigenfunction configurations of s‐, p‐, and d‐Slater orbitals. Two procedures of optimization of the orbital exponents have been investigated. First, CI(SD) calculations including few types of configurations and full optimization of the orbital exponents led to the energy ?24.63704575 a.u. Second, full‐CI (FCI) calculations including a large number of configuration types using a fixed set of orbital exponents for all configurations gave ?24.63405222 a.u. using the basis [4s3p2d] and 2157 configurations, and to an improved result of ?24.64013999 a.u. for 3957 configurations and a [5s4p3d] basis. This last result is better than earlier calculations of Schaefer and Harris (Phys Rev 1968, 167, 67), and compares well with the recent ones from Froese Fischer and Bunge (personal communication). In addition, using the same wave functions, CI calculations of the boron isoelectronic ion C⁺ have been performed obtaining an energy of ?37.41027598 a.u. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

First‐order correlation‐kinetic contribution to Kohn–Sham exchange charge density function in atoms,using quantal density functional theory approach

Using the static exchange‐correlation charge density concept, the total integrated exchange‐charge density function is calculated within the nonrelativistic spin‐restricted exchange‐only (i) optimized effective potential model, and (ii) nonvariational local potential derived from the exchange‐only work potential within the quantal density functional theory, for the ground‐state isoelectronic series: Ga⁺, Zn, Cu^?; In⁺, Cd, Ag^?; and Tl⁺, Hg, Au^?. The difference between the exchange charge density function derived from these potentials is employed to evaluate the first‐order correlation‐kinetic contribution to the integrated exchange charge density. This contribution is found to be important for both the intra‐ and inter‐shell regions. Screening effects on the contribution due to the nd¹⁰ (n = 3–5) subshells are discussed through comparisons with similar calculations on Ca, Sr, and Ba, wherein nd¹⁰ electrons are absent. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Molecular orbital studies on small molecules using H2+-type elliptical basis orbitals. Application to H2+, H2, He2++ and H3+

H₂⁺-type elliptical orbitals are defined in Section 1. These orbitals, which in elliptical coordinates involve a factor (1 + ξ)^σ, are employed in variational calculations on the ground states of H₂⁺ and H₂ (Sections 2 and 3). Various choices of σ are explored for H₂⁺, while two choices are used for H₂ : the “boundary condition” (Equation 6) and the “cusp condition” (Equation 9) values. Variational energies are calculated and compared to the results of similar calculations. Section 3 concludes by employing the H₂⁺-type orbitals in LCETO-MO-SCF calculations on the ground states of H₂ and He₂⁺⁺. For both molecules a four-function basis set with two (nonlinear) variational parameters yields more than 99% of the Hartree-Fock limit. Section 4 deals with LCETO-MO-SCF calculations on triangular H₃⁺. Three four-function basis sets are used, and the best energy is -1.2306 a.u., which is in reasonable agreement with the Hartree-Fock limit, -1.2999 a.u. Our best basis set is a four-term two-center expansion of the wave function with only one nonlinear variational parameter. Section 5 concludes the paper with a summary of the methods used to evaluate the integrals which arise in SCF calculations in the H₂⁺-type elliptical orbital basis.     

High precision variational calculations for the Born-Oppenheimer energies of the ground state of the hydrogen molecule

Born-Oppenheimer approximation Hylleraas variational calculations with up to 7034 expansion terms are reported for the 1sigma(g)+ ground state of neutral hydrogen at various internuclear distances. The nonrelativistic energy is calculated to be -1.174 475 714 220(1) hartree at R = 1.4 bohr, which is four orders of magnitude better than the best previous Hylleraas calculation, that of Wolniewicz [J. Chem. Phys. 103, 1792 (1995)]. This result agrees well with the best previous variational energy, -1.174 475 714 216 hartree, of Cencek (personal communication), obtained using explicitly correlated Gaussians (ECGs) [Cencek and Rychlewski, J. Chem. Phys. 98, 1252 (1993); Cencek et al., ibid. 95, 2572 (1995); Rychlewski, Adv. Quantum Chem. 31, 173 (1998)]. The uncertainty in our result is also discussed. The nonrelativistic energy is calculated to be -1.174 475 931 399(1) hartree at the equilibrium R = 1.4011 bohr distance. This result also agrees well with the best previous variational energy, -1.174 475 931 389 hartree, of Cencek and Rychlewski [Rychlewski, Handbook of Molecular Physics and Quantum Chemistry, edited by S. Wilson (Wiley, New York, 2003), Vol. 2, pp. 199-218; Rychlewski, Explicitly Correlated Wave Functions in Chemistry and Physics Theory and Applications, edited by J. Rychlewski (Kluwer Academic, Dordrecht, 2003), pp. 91-147.], obtained using ECGs.     

On the behaviour of the wave function for the 23S state of the two-electron atom in the region where both electron–nuclear distances are small

The expansion of the wave function for the 2³S state of the two-electron atom in the neighbourhood of the singularity at r₁ = r₂ = 0 is considered. The restrictions imposed on the variational functions by this expansion are discussed. For the 2³S state of He, Li⁺, N⁵⁺ the behaviour of the variational function based on the Fock expansion in the neighbourhood of this singularity is investigated. The agreement of the variational coefficients with the theoretical coefficients is satisfactory. The calculated values of E and 〈δ(r₂)〉 for He, Li⁺, N⁵⁺ are given.     

Improved Hylleraas calculations for ground state energies of lithium ISO–electronic sequence

The ²S ground state of lithium iso–electronic sequence is calculated by the use of Hylleraas-type wave functions. A 92 term one-spin wave function was used for lithium atom calculations. The energy obtained was ?7.478031 a.u. as compared with the previous best value of ?7.478025 a.u. calculated by Larsson. In addition, improved energies for Z = 4 to 8 were calculated by the use of 60 term wave functions. This work thus provides the lowest ab initio ground state energies for lithium sequence to date.     

Long- and intermediate-range interaction in three lowest sigma states of the HeH+ ion

Accurate variational energies have been calculated for three lowest sigma states of the HeH⁺ ion. This includes the ground state (5 ≤ R ≤ 9 a.u.) which dissociates into He + H⁺, as well as the A ¹Σ⁺ state (4 ≤ R ≤ 10) and the a ³Σ⁺ state (3 ≤ R ≤ 10) which both dissociate into He⁺ + H. The variational results are compared with those obtained using a perturbation theory expansion.     

Two-Dimensional fully numerical solutions of molecular Schrödinger equations. II. Solution of the Poisson equation and results for singlet states of H2 and HeH+

Two-dimensional fully numerical solutions of the Hartree–Fock problem are reported for the singlet ground states of H^?, He, H₂, and HeH⁺. The H₂ energy at R = 1.4 a.u. is ?1.13362957 a.u.     

ELECTRON TRANSFER FROM PHOTOEXCITED SINGLET AND TRIPLET BACTERIOPHEOPHYTIN—II. THEORETICAL

Abstract— A previous paper showed that collision of the first excited singlet state of bacteriopheophytin (Bph*) and p-benzoquinone (Q) returns Bph* to the ground state; however, excited triplet (Bph⁺) and quinone on collision produce the radical ions, (Bph⁺) and (Q^?). This paer rationalizes these findings by first estimating the half cell potentials Bph⁺/Bph* and Bph⁺/Bph^T, the energy for the various collision complexes, and the energy of the charge separated ions Bph⁺+ Q? and then estimating the rates for conversion among these various states. Thus it is estimated that the complexes [Bph*Q] or [Bph^TQ], live ?5 ps before dissociating. This is long enough for electron transfer to occur, producing the singlet and triplet charge transfer complexes, [Bph⁺Q?]^S or [Bph⁺Q?]^T, either of which could separate to Bph⁺+ Q? in ?230ps. In the singlet case, quenching by reverse charge transfer [Bph⁺Q?]^S→[Bph Q] occurs more rapidly than ion separation; however, the analogous triplet process, [Bph⁺Q?]^T→ [Bph Q], is spin forbidden, so that ion separation competes successfully with quenching. Spin scrambling, [Bph⁺Q?]^S? [Bph⁺Q?]^T, is estimated to be slow, as this explanation requires. In the bacterial photosynthetic reaction center, the initial electron transfer from an excited singlet state of the bacteriochlorophyll dimer complex (BB)* to bacteriopheophytin, giving [(BB⁺)(Bph?)]^S, successfully leads to ion separated species (i) because reverse charge transfer [(BB⁺)(Bph?)]^S→ [(BB)(Bph)] is slowed by a fairly large Franck-Condon energy, ΔE? lev, which is difficult to convert from electronic to vibrational degrees of freedom and (ii) because of the rapid subsequent electron transfer from (Bph⁺) to another acceptor X.     

Variational Monte Carlo calculations for some cations and anions of the first‐row atoms using explicitly correlated wave functions

The variational Monte Carlo method is applied to calculate ground‐state energies of some cations and anions of the first‐row atoms. Accurate values providing between 80 and 90% of the correlation energy are obtained. Explicitly correlated wave functions including up to 42 variational parameters are used. The nondynamic correlation due to the 2s ? 2p near degeneracy effect is included by using a multideterminant wave function. The variational free parameters have been fixed by minimizing the energy that has shown to be a more convenient functional than the variance of the local energy, which is the most commonly employed method in variational Monte Carlo calculations. The energies obtained improve previous works using similar wave functions. © 2002 Wiley Periodicals, Inc.; DOI 10.1002/qua.10125     

Controlling Helical Chirality in Atrane Structures: Solvent‐Dependent Chirality Sense in Hemicryptophane‐Oxidovanadium(V) Complexes

The diastereomeric hemicryptophane oxidovanadium(V) complexes (P)‐(S,S,S)‐ 3 and (M)‐(S,S,S)‐ 4 have been synthesized. ¹H and ⁵¹V NMR spectra in solution are consistent with the formation of Λ and Δ forms of the propeller‐like vanatrane moiety, leading to two diastereomeric conformers for each complex: that is, (P)‐(S,S,S‐Λ)‐ 3 /(P)‐(S,S,S‐Δ)‐ 3 and (M)‐(S,S,S‐Λ)‐ 4 /(M)‐(S,S,S‐Δ)‐ 4 . The Λ/Δ ratio is rather temperature‐insensitive but strongly dependent on the solvent (the de of (M)‐(S,S,S)‐ 4 changes from 0 in benzene to 92 % in DMSO). The solvent therefore controls the preferential clockwise or anticlockwise orientation of the propeller‐like atrane unit. The energy barriers for the Λ?Δ equilibrium were determined by NMR experiments, and the highest ΔG^≠ value (103.7 kJ mol^?1) was obtained for (P)‐(S,S,S)‐ 3 , much higher than those reported for other atrane derivatives. This is attributed to the constraints arising from the cage structure. Determination of the activation parameters provides evidence for a concerted, rather than a stepwise, interconversion mechanism with entropies (ΔS^≠) of ?243 and ?272 J mol^?1 K^?1 for (P)‐(S,S,S)‐ 3 and (M)‐(S,S,S)‐ 4 , respectively. The molecular structure of the (P)‐(S,S,S‐Λ)‐ 3 isomer was solved by X‐ray diffraction and shows a distorted structure with one of the linkers located in the CTV cavity. Complementary quantum chemical calculations were carried out to obtain the energy‐minimized structures of (P)‐(S,S,S)‐ 3 and (M)‐(S,S,S)‐ 4 . Our density functional theory calculations suggest that the (P)‐(S,S,S‐Λ)‐ 3 is favored, in agreement with experimental data. For the M series, a similar strategy was used to extract molecular structures and relative energies. As in the case of the P diastereomer, the Λ form dominates over the Δ one.     

Hydrogenlike atoms in uniformly charged sphere model of atomic nucleus. II. Application of basis-set expansion method

The basis-set expansion method has been applied to the nonrelativistic and relativistic ground states of one-electron atoms H, Ar¹⁷⁺, Xe⁵³⁺, Hg⁷⁹⁺, and U⁹¹⁺ in the uniformly charged sphere (UCS ) and the point-change (PC ) models of atomic nucleus, using the Gaussian-type basis functions. The energies and the radial expectation values, especially 1/r², converges faster in the UCS model than in the PC model. In the PC model, larger values of the exponent parameters of the basis functions are required both in the nonrelativistic and the relativistic calculations. Even in the UCS model, the larger values of the exponent parameters are needed in the relativistic calculations.     

Analytical gradients for Singer's multicenter n‐electron explicitly correlated Gaussians

Analytical gradients for Singer's basis of n‐electron multicenter explicitly correlated Gaussian functions are derived and implemented to variationally optimize the energy and wave function of molecular systems within the Born–Oppenheimer approximation. Wave functions are optimized with respect to (½n(n+1)+3n) nonlinear variational parameters and one linear coefficient per term in the basis set. Preliminary results for the ground states of H₃⁺ and H₃ suggest that the method can be more flexible and can achieve lower energies than previously reported calculations. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 151–159, 2001     

Ring‐opening polymerization of ϵ‐caprolactone by iodine charge‐transfer complex

The ring‐opening polymerization of ?‐caprolactone (?‐CL) catalyzed by iodine (I₂) was studied. The formation of a charge‐transfer complex (CTC) among triiodide, I, and ?‐CL was confirmed with ultraviolet–visible spectroscopy. The monomer ?‐CL was polymerized in bulk using I₂ as a catalyst to form the polyester having apparent weight‐average molecular weights of 35,900 and 45,500 at polymerization temperatures of 25 and 70 °C, respectively. The reactivity of both, ?‐CL monomer and ?‐CL:I₂ CTC, was interpreted by means of the potential energy surfaces determined by semiempirical computations (MNDO‐d). The results suggest that the formation of the ?‐CL:I₂ CTC leads to the ring opening of the ?‐CL structure with the lactone protonation and the formation of a highly polarized polymerization precursor (?‐CL)⁺. The band gaps approximated from an extrapolation of the oligomeric polycaprolactone (PCL) structures were computed. With semiempirical quantum chemical calculations, geometries and charge distributions of the protonated polymerization precursor (?‐CL)⁺ were obtained. The calculated band gap (highest occupied molecular orbit/lowest unoccupied molecular orbit differences) agrees with the experiment. The analysis of the oligomeric PCL isosurfaces indicate the existence of a weakly lone pair character of the C?O and C? O bonds suggesting a ?‐CL ring‐opening specificity. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 714–722, 2002     

Near‐IR Absorbing Ruthenium Complexes of Non‐Innocent 6,12‐Di(pyridin‐2‐yl)indolo[3,2‐b]carbazole: Variation as a Function of Co‐Ligands

This article deals with the hitherto unexplored metal complexes of deprotonated 6,12‐di(pyridin‐2‐yl)‐5,11‐dihydroindolo[3,2‐b]carbazole (H₂L). The synthesis and structural, optical, electrochemical characterization of dimeric [{Ru^III(acac)₂}₂(μ‐L^.?)]ClO₄ ([ 1 ]ClO₄, S=1/2), [{Ru^II(bpy)₂}₂(μ‐L^.?)](ClO₄)₃ ([ 2 ](ClO₄)₃, S=1/2), [{Ru^II(pap)₂}₂(μ‐L^2?)](ClO₄)₂ ([ 4 ](ClO₄)₂, S=0), and monomeric [(bpy)₂Ru^II(HL^?)]ClO₄ ([ 3 ]ClO₄, S=0), [(pap)₂Ru^II(HL^?)]ClO₄ ([ 5 ]ClO₄, S=0) (acac=σ‐donating acetylacetonate, bpy=moderately π‐accepting 2,2’‐bipyridine, pap=strongly π‐accepting 2‐phenylazopyridine) are reported. The radical and dianionic states of deprotonated L in isolated dimeric 1 ⁺/ 2 ³⁺ and 4 ²⁺, respectively, could be attributed to the varying electronic features of the ancillary (acac, bpy, and pap) ligands, as was reflected in their redox potentials. Perturbation of the energy level of the deprotonated L or HL upon coordination with {Ru(acac)₂}, {Ru(bpy)₂}, or {Ru(pap)₂} led to the smaller energy gap in the frontier molecular orbitals (FMO), resulting in bathochromically shifted NIR absorption bands (800–2000 nm) in the accessible redox states of the complexes, which varied to some extent as a function of the ancillary ligands. Spectroelectrochemical (UV/Vis/NIR, EPR) studies along with DFT/TD‐DFT calculations revealed (i) involvement of deprotonated L or HL in the oxidation processes owing to its redox non‐innocent potential and (ii) metal (Ru^III/Ru^II) or bpy/pap dominated reduction processes in 1 ⁺ or 2 ²⁺/ 3 ⁺/ 4 ²⁺/ 5 ⁺, respectively.     

Solvent Migration in Microhydrated Aromatic Aggregates: Ionization‐Induced Site Switching in the 4‐Aminobenzonitrile–Water Cluster

The dependence of the preferred microhydration sites of 4‐aminobenzonitrile (4ABN) on electronic excitation and ionization is determined through IR spectroscopy of its clusters with water (W) in a supersonic expansion and through quantum chemical calculations. IR spectra of neutral 4ABN and two isomers of its hydrogen‐bonded (H‐bonded) 4ABN–W complexes are obtained in the ground and first excited singlet states (S₀, S₁) through IR depletion spectroscopy associated with resonance‐enhanced multiphoton ionization. Spectral analysis reveals that electronic excitation does not change the H‐bonding motif of each isomer, that is, H₂O binding either to the CN or the NH site of 4ABN, denoted as 4ABN–W(CN) and 4ABN–W(NH), respectively. The IR spectra of 4ABN⁺–W in the doublet cation ground electronic state (D₀) are measured by generating them either in an electron ionization source (EI‐IR) or through resonant multiphoton ionization (REMPI‐IR). The EI‐IR spectrum shows only transitions of the most stable isomer of the cation, which is assigned to 4ABN⁺–W(NH). The REMPI‐IR spectrum obtained through isomer‐selective resonant photoionization of 4ABN–W(NH) is essentially the same as the EI‐IR spectrum. The REMPI‐IR spectrum obtained by ionizing 4ABN–W(CN) is also similar to that of the 4ABN⁺–W(NH) isomer, but differs from that calculated for 4ABN⁺–W(CN), indicating that the H₂O ligand migrates from the CN to the NH site upon ionization with a yield of 100 %. The mechanism of this CN→NH site‐switching reaction is discussed in the light of the calculated potential energy surface and the role of intracluster vibrational energy redistribution.