Exact Radial Solution of the Non-relativistic Schr?dinger Equation for the Helium Atom with the Potential Harmonic Method

We proposed a simple potential harmonic(PH) scheme for calculating the non-relativistic radial correlation energies of atomic systems.The scheme was applied to the low-lying n1S(n=1,2) and n3S(n=2,3) states of the helium atom.The results exhibit a very stable convergence characterization in both the angular and radial directions with PH and generalized Laguerre functions(GLF) respectively,even though the method is non-variational one.The ninth significant figure of the non-relativistic radial energy(NRE) calculated for the ground state exactly agrees with that of the most accurate literature data from the modified configuration interaction method.The convergent NRE′s for the excited states 21S,23S and 33S with the similar accuracy were also obtained.     

PHGLF calculations of heliumlike three-body systems

By expanding the atomic wave function in potential harmonics (PH ) and generalized Laguerre functions (GLF ), we derived the recurrence relation of the expansion coefficients and then performed calculations for four He-like three-body systems. In comparison with the complete set calculation (HHGLF ), we find that the errors in eigenenergies introduced by the PHGLF scheme are approximately 0.00033, 0.00025, 0.00022, and 0.00021 au for the ground-state H⁻, He, Li⁺, and Be²⁺ systems, respectively, and that for excited states the error is slightly higher and decreases with the grand angular momentum. The present results indicate that the PHGLF scheme is such an efficient means to reduce the degeneracy of the hyperspherical harmonics (HH ) for three-body problems that it can give more satisfactory results than can the HHGLF method with much smaller eigenmatrices. © 1995 John Wiley & Sons, Inc.     

超球谐展开的收敛行为He原子的^3S态

利用相关函数－超球谐－广义Ｌａｇｕｅｒｒｅ函数方法，研究Ｈｅ原子＾３Ｓ态波函数向超球谐函数展开的收敛行为。截止于ｌ＝０，１，２的超球谐函数给出的本征能分别与组态相互作用的ｓ－，ｓｐ－，ｓｐｄ－极限能一致。仅用４４个超球谐函数，便得到了与精确的ＨｙｌｌｅｒａａｓＣＩ变分能量小数点后第５位的２＾３Ｓ，３＾３Ｓ的４＾３Ｓ态本征能吻合。     

The eigenenergies for ~3S excited states of the helium atom with CFPHGLF method

The matrix elements of the correlation function between symmetric potential harmonics were first simplified into the analytical summation of the grand angular momentum. The correlation-function potential-harmonic and generalized Laguerre function method (CFPHGLF) proposed by us recently was then applied to directly solve the Schrodinger equation for n3S(n=2-5) excited states of the helium atom. With only 12 PHs, the convergent eigenenergies of 23S, 33S, 43S and 53S states were 2.17427, 2.06849, 2.03644, 2.02257 Eh, respectively. The errors only were 0.00096, 0.00020, 0.00007, 0.00005 Eh, when compared with the exact Hylleraas variational results respectively.     

Optimizing the structures of minimum and transition state on the free energy surface

Presented here is the application of a scheme for optimizing the structures of minima and transition states on the free energy surface (FES) for a path along a fixed reaction coordinate with the aid of ab initio molecular dynamics (AIMD) simulation. In the direction of the reaction coordinate, the values corresponding to the stationary points were optimized using the quasi-Newton method, in which the gradient of the free energy along the reaction coordinate was obtained by a constraint AIMD method, and the Bofill Hessian update scheme was used. The equilibrium values for the other directions were taken as the corresponding averages in the dynamic simulation. This scheme was applied to several elementary bimolecular addition reactions: (A) BH(3) + H(2)O --> H(2)O.BH(3); (B) BF(3) + NH(3) --> FB(3).NH(3); (C) SO(3) + NH(3) --> O(3)S.NH(3); (D) C(2)H(4) + CCl(2) --> H(4)C(2).CCl(2); (E) Ni(NH(2))(2) + PH(3) --> (NH(2))(2)Ni.PH(3); (F) W(CO)(5) + CO --> W(CO)(6). For reactions A, B, C, and F, no transition state (TS) exists on the potential energy surface (PES). However there is a TS on the FES. This stems from the curvature difference of the PES and -TDeltaS as a function of the reaction coordinate. For all reactions, it is found that the TS shifts toward the complexation product with increasing temperature because of the curvature increase of -TDeltaS. The equilibrium bond distances for the inactive coordinates perpendicular to the reaction coordinate always increase with temperature, which is due to the thermal excitation and anharmonicity of the PES.     

Ab initio calculation of the NH(3sigma-)-NH(3sigma-) interaction potentials in the quintet, triplet, and singlet states

We present the ab initio potential-energy surfaces of the NH-NH complex that correlate with two NH molecules in their 3sigma- electronic ground state. Three distinct potential-energy surfaces, split by exchange interactions, correspond to the coupling of the S(A) = 1 and S(B) = 1 electronic spins of the monomers to dimer states with S = 0, 1, and 2. Exploratory calculations on the quintet (S = 2), triplet (S = 1), and singlet (S = 0) states and their exchange splittings were performed with the valence bond self-consistent-field method that explicitly accounts for the nonorthogonality of the orbitals on different monomers. The potential surface of the quintet state, which can be described by a single Slater determinant reference function, was calculated at the coupled cluster level with single and double excitations and noniterative treatment of the triples. The triplet and singlet states require multiconfiguration reference wave functions and the exchange splittings between the three potential surfaces were calculated with the complete active space self-consistent-field method supplemented with perturbative configuration interaction calculations of second and third orders. Full potential-energy surfaces were computed as a function of the four intermolecular Jacobi coordinates, with an aug-cc-pVTZ basis on the N and H atoms and bond functions at the midpoint of the intermolecular vector R. An analytical representation of these potentials was given by expanding their dependence on the molecular orientations in coupled spherical harmonics, and representing the dependence of the expansion coefficients on the intermolecular distance R by the reproducing kernel Hilbert space method. The quintet surface has a van der Waals minimum of depth D(e) = 675 cm(-1) at R(e) = 6.6a0 for a linear geometry with the two NH electric dipoles aligned. The singlet and triplet surfaces show similar, slightly deeper, van der Waals wells, but when R is decreased the weakly bound NH dimer with S = 0 and S = 1 converts into the chemically bound N2H2 diimide (also called diazene) molecule with only a small energy barrier to overcome.     

超球坐标下Li原子^4S态Schrodinger方程的直接解

以九维超球谐为投影函数, 构造置换群S3一维不可约表示的全反称基函数。应用HHGLF和CFHHGLF方法直接求解Li原子^4S态的Schrodinger方程。由HHGLF方法得到的本征能量与精确变分数值相差较远; CFHHGLF方法给出了与变分值相近的结果。     

Fast hydrogen elimination from the [Ru(PH3)3(CO)(H)2] and [Ru(PH3)4(H)2] complexes in the first singlet excited states: a diabatic quantum dynamics study

The photodissociation dynamics of [Ru(PH3)3(CO)(H)2] and cis-[Ru(PH3)4(H)2] is theoretically analyzed in the lowest two excited singlet states. Energies obtained through electronic density functional theory calculations that use the time-dependent formalism are fitted to analytical reduced two-dimensional potential energy surfaces (2D-PES). The metal-H2 (R) and H-H (r) distances are the variables of these 2D-PES, the rest of the parameters being kept frozen at the values of the minimum energy structure in the ground electronic state. The time evolution in these 2D-PES is exactly followed by means of a fast Fourier transform algorithm applied to solve the time-dependent Schr?dinger equation. A simple diabatization scheme is devised to take into account the probability of transitions between both excited states. The quantum dynamics results point out that photoelimination is almost inexistent if the H2 fragment is to be expelled without further rearrangement of the rest of the complex. Conversely, when the geometries of the complex are optimized by keeping r and R frozen at the hydrogen elimination barrier coordinates, the new 2D-PES so obtained are highly dissociative, the H2 fragment being expelled in less than 100 fs. Finally the picture of the whole reaction that emerges from our theoretical results is described and the main differences between both complexes are examined.     

Excitation of ions by high-harmonic frequency components in Paul and Penning traps and ion guides. I. Selective simultaneous dipolar excitation of high charge states with clipped sinusoidal and non-harmonic waveforms in a linear quadrupole ion guide

This article presents a method for simultaneous excitation of multiple high charge states of a molecular ion in Paul or Penning trap. Using a linear quadrupolar ion guide we validate the method by using a variety of time- domain excitation waveforms with high harmonics of the first charge state's resonant frequency. The proposed way of inducing harmonics is the deliberate distortion of the excitation waveform from its sinusoidal form. In order to facilitate interference of the harmonics, a superposition of two sinusoids different by a frequency factor of two is used. The simplest form of distortion - amplitude restriction - of such waveform produces interference of the harmonics and results in selective excitation of charge states. Multiple protonation states of melittin were used as a model in this study.     

包括电子间相关势谐函数的CFPHGLF方法

包括电子间相关势谐函数的ＣＦＰＨＧＬＦ方法王沂轩,邓从豪（山东大学化学院,济南,２５０１００）关键词超球坐标,势谐函数,快速收敛前文［１,２］吸收相关函数（ＣＦ）的思想,与势谐函数－广义Ｌａｇｕｅｒｒｅ函数（ＧＬＦ）方法（ＰＨＧＬＦ）［３,４］相结合...     

Calculation of excited singlet and triplet state polarizabilities using the CNDO/S CI method. An application to naphthalene

A procedure is outlined for the calculation of molecular static electric polarizabilities in excited singlet and triplet states using the ”finite perturbation theory“ in conjunction with the CNDO/S CI method. Numerical results for the ground and the three lowest excited singlet and triplet states of naphthalene are presented. It turns out that the generalized Hellmann-Feynman theorem is approximately valid for the CNDO/S CI wavefunctions and that triplet and singlet state polarizabilities in states of the same symmetry may strongly differ.     

A note on atomic density

In atomic systems, electron density has a simple finite expansion in spherical harmonics times radial factors. The difficulties in the calculation of some radial factors are illustrated in the low‐lying states of the carbon atom. Single‐particle methods such as Hartree–Fock and approximate density functional theory cannot ensure the correct expansion of the density in spherical harmonics. Wave‐function methods are appropriate but, as some expansion terms are entirely due to correlation, these methods only will give correct results for high‐quality variational functions. Using full‐configuration integration (CI), all the terms predicted by the theory appear and are not negligible but the convergence of the term due to correlation toward its correct value is uncertain even for very large CI spaces. © 2012 Wiley Periodicals, Inc.     

Performance of the general-model-space state-universal coupled-cluster method

The capabilities of the recently developed multireference, general-model-space (GMS), state-universal (SU) coupled-cluster (CC) method have been extended in order to enable the handling of any excited state that represents a single (S) or a double (D) excitation relative to the ground state. A series of calculations concerning the ground and excited states of the CH(+), HF, F(2), H(2)O, NH(2), and CH(2) molecules were carried out so as to assess the performance of the GMS SU CCSD method. For diatomics we have computed the entire potential energy curves, while for triatomics we have focused on vertical excitation energies. We demonstrate how a systematic enlargement of the model space enables a consideration of a larger and larger number of excited states. A comparison of the CC and full configuration interaction or large-scale CI results enables an assessment of the accuracy and reliability of the GMS SU CCSD method within a given basis set. In all cases very good results have been obtained, including highly excited states and those having a doubly-excited character.     

Optimizing conical intersections of solvated molecules: the combined spin-flip density functional theory/effective fragment potential method

Solvent effects on a potential energy surface crossing are investigated by optimizing a conical intersection (CI) in solution. To this end, the analytic energy gradient has been derived and implemented for the collinear spin-flip density functional theory (SFDFT) combined with the effective fragment potential (EFP) solvent model. The new method is applied to the azomethane-water cluster and the chromophore of green fluorescent protein in aqueous solution. These applications illustrate not only dramatic changes in the CI geometries but also strong stabilization of the CI in a polar solvent. Furthermore, the CI geometries obtained by the hybrid SFDFT/EFP scheme reproduce those by the full SFDFT, indicating that the SFDFT/EFP method is an efficient and promising approach for understanding nonadiabatic processes in solution.     

Numerical Convergence of the Sinc Discrete Variable Representation for Solving Molecular Vibrational States with a Conical Intersection in Adiabatic Representation

Within the Born-Oppenheimer (BO) approximation, nuclear motions of a molecule are often envisioned to occur on an adiabatic potential energy surface (PES). However, this single PES picture should be reconsidered if a conical intersection (CI) is present, although the energy is well below the CI. The presence of the CI results in two additional terms in the nuclear Hamiltonian in the adiabatic presentation, i.e., the diagonal BO correction (DBOC) and the geometric phase (GP), which are divergent at the CI. At the same time, there are cusps in the adiabatic PESs. Thus usually it is regarded that there is numerical difficulty in a quantum dynamics calculation for treating CI in the adiabatic representation. A popular numerical method in nuclear quantum dynamics calculations is the Sinc discrete variable representation (DVR) method. We examine the numerical accuracy of the Sinc DVR method for solving the Schr?dinger equation of a two dimensional model of two electronic states with a CI in both the adiabatic and diabatic representation. The results suggest that the Sinc DVR method is capable of giving reliable results in the adiabatic representation with usual density of the grid points, without special treatment of the divergence of the DBOC and the GP. The numerical uncertainty is not worse than that after the introduction of an arbitrary vector potential for accounting the GP, whose accurate form usually is not easy to obtain.     

A comparative calculation on excited state energies of some conjugated hydrocarbons

The energies of the single-configuration lowest π – π* singlet and triplet states of some conjugated hydrocarbons have been calculated by the MC-SCF method using the conjugate-gradient technique of minimisation. The results are compared with those calculated by other methods currently in use, like (a) single-configuration calculation with V_N?1 potential for virtual orbitals; (b) CI calculation involving singly excited states; and (c) TDHF method. It has been concluded that the results for the MC-SCF method are very good, considering that only a single open-shell configuration is involved.     

Photodissociation of the phosphine-substituted transition metal carbonyl complexes Cr(CO)(5)L and Fe(CO)(4)L: a theoretical study

The photochemistry of the phosphine-substituted transition metal carbonyl complexes Cr(CO)(5)PH(3) and ax-Fe(CO)(4)PH(3) is studied with time-dependent DFT theory to explore the propensity of the excited molecules to expel their ligands. The influence of the PH(3) ligand on the properties of these complexes is compared with the photodissociation behavior of the binary carbonyl complexes Cr(CO)(6) and Fe(CO)(5). The lowest excited states of Cr(CO)(5)PH(3) are metal-to-ligand charge transfer (MLCT) states, of which the first three are repulsive for PH(3) but modestly bonding for the axial and equatorial CO ligands. The repulsive nature is due to mixing of the initial MLCT state with a ligand field (LF) state. A barrier is encountered along the dissociation coordinate if the avoided crossing between these states occurs beyond the equilibrium distance. This is the case for expulsion of CO but not for the PH(3) group as the avoided state crossing occurs within the equilibrium Cr-P distance. The lowest excited state of ax-Fe(CO)(4)PH(3) is a LF state that is repulsive for both PH(3) and the axial CO. Excited-state quantum dynamics calculations for this state show a branching ratio of 99 to 1 for expulsion of the axial phosphine ligand over an axial CO ligand. The nature of the phosphorus ligand in these Cr and Fe complexes is only of modest importance. Complexes containing the three-membered phosphirane or unsaturated phosphirene rings have dissociation curves for their lowest excited states that are similar to those having a PH(3) ligand. Analysis of their ground-state Cr-P bond properties in conjunction with frontier orbital arguments indicate these small heterocyclic groups to differ from the PH(3) group mainly by their enhanced sigma-donating ability. All calculations indicate that the excited Cr(CO)(5)L and Fe(CO)(4)L molecules (L = PH(3), PC(2)H(5), and PC(2)H(3)) prefer dissociation of their phosphorus substituent over that of an CO ligand. This suggests that the photochemical approach may be a viable complement to the ligand exchange and redox methods that are currently employed to demetalate transition metal complexed organophosphorus compounds.     

Distinct photophysics of the isomers of B18H22 explained

The photophysics of the two isomers of octadecaborane(22), anti- and syn-B(18)H(22), have been studied by UV-vis spectroscopic techniques and theoretical computational methods. In air-saturated hexane, anti-B(18)H(22) shows fluorescence with a high quantum yield, Φ(F) = 0.97, and singlet oxygen O(2)((1)Δ(g)) production (Φ(Δ) ～ 0.008). Conversely, isomer syn-B(18)H(22) shows no measurable fluorescence, instead displaying much faster, picosecond nonradiative decay of excited singlet states. Computed potential energy hypersurfaces (PEHs) for both isomers rationalize these data, pointing to a deep S(1) minimum for anti-B(18)H(22) and a conical intersection (CI) between its S(0) and S(1) states that lies 0.51 eV higher in energy. Such an energy barrier to nonradiative relaxation is not present in the PEH of syn-B(18)H(22), and the system therefore has sufficient initial energy on excitation to reach the (S(0)/S(1)) CI and to then decay to the ground state without fluorescence. The computational analysis of the geometries at stationary points along the PEH of both isomers shows that the determining factor for the dissimilar photophysics of anti- and syn-B(18)H(22) may be due to the significant differences in the geometrical rearrangements at their respective conical intersections. Thus, the syn isomer shows one very large, B-B elongation of 1.2 ? from 1.8 ? in the ground state to 3.0 ? at the CI, whereas the anti isomer shows smaller elongations (below 1 ?) in several B-B connectivities at its (S(0)/S(1))(CI). The absorbed energy in S(1) for the anti-B(18)H(22) is therefore redistributed vibrationally into several regions of the molecule rather than almost completely into a single vibrational mode as in the case for the syn isomer. The consequent prolonged S(1) lifetime for the anti isomer allows for relaxation via fluorescence.     

New insights in the formation of thioxophosphine: a quantum chemical study

The investigation of the thioxophosphine (PS) formation from different reaction paths is successfully performed and presented in this paper. The PH(3)+SH(1) reaction is likely to yield the intermediates PH(2) (2)+H(2)S through an energy barrier of 2.8 kcal mol(-1). However, the next step is the H(2)PS(2) formation, which has a too high energy barrier, 52.6 kcal mol(-1). The PH(3)+S(1) reaction path is the likely source of the HPS(1) molecule. The other possibilities are the PH(1)+H(2)S, PH(2) (2)+SH(1), and PH(3)+H(2)S reactions, but they are spin forbidden and energetically unfavorable for the HPS(1) and PSH(1) formations. On the other hand, the PS(2) formation is more likely to happen by the PH(1)+SH(1) reaction. The PH(2) (2)+S(1), PH(3)+SH(1), P(2)+H(2)S, and P(4)+H(2)S reactions are also favorable in terms of energetics; however, these reactions are spin forbidden. The chemical mechanism for the PS(2) formation is now presented in more details, which is of great importance in the atmosphere of Jupiter and Saturn, and in interstellar medium.