Nonlinear Variational Parameters Research Articles

Low-lying S2 states of the singly charged carbon ion

In this work, we report benchmark variational calculations of the five lowest doublet $S$-states of the ${\mathrm{C}}^{+}$ ion. The wave functions of this five-electron system are expanded in terms of 16 000 all-particle explicitly correlated Gaussians (ECGs) whose nonlinear variational parameters are subject to extensive optimization. The motion of the finite-mass nucleus is explicitly included in the Hamiltonian, while relativistic corrections to the energy levels are computed in the framework of the perturbation theory. Lowest-order quantum electrodynamics (QED) corrections are also estimated. The results obtained for the energy levels enable the determination of transition frequencies with the accuracy that approaches the available experimental data and may open up avenues for future determination of nuclear charge radii of carbon isotopes.

Two-Parameter Harmonic Oscillator Expansion Method for Calculating Non-Relativistic Ground State Energy of Coulomb Three-Particle Systems with Two Identical Particles

The variational method in oscillator representation with individual parameters for each Jacobi coordinate is applied to the non-relativistic calculation of the ground state energy of a number of three-particle Coulomb systems, consisting of two identical particles and a different one. The accuracy and convergence rate of the calculations in the constructed oscillator basis are studied up to a total of 28 oscillator quanta. The results are compared with those of the traditional approach using only one such nonlinear variational parameter. The method with individual parameters for Jacobi coordinates is found to possess a number of advantages as compared to the traditional approach.

Polarizability of negatively charged helium‐like ions interacting with Coulomb and screened Coulomb potentials

AbstractThe dipole and quadrupole polarizabilities (both static and dynamic) of negatively charged helium‐like ions are investigated. The mass dependence of the polarizability is studied by changing the mass of the positively charged particle from one unit of electron mass to infinitely heavy. The calculations are carried out in the framework of the pseudostate summation method using exponential correlated wave functions having pseudorandomly generated nonlinear variational parameters. The dipole and quadrupole polarizabilities in terms of frequency and nuclear mass are reported for the first time. The effect of screened Coulomb potentials on the polarizabilities of D–, T–, 1H–,Pi–, Mu–, and Ps– are also presented.

Anharmonic oscillator and the optimized basis expansion

We introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one-dimensional anharmonic oscillators. We present several methods of analytically fixing the nonlinear variational parameter specified by the domain of the trigonometric basis functions. We show that the optimized parameter enables us to determine the energy spectrum to an arbitrary accuracy. Also, using the harmonic oscillator basis functions, we indicate that the resulting optimal frequency agrees with the one obtained by the principle of the minimal sensitivity.

‘tripleint_cc’: A program for 2-centre variational leptonic Coulomb potential matrix elements using Hylleraas-type trial functions, with a performance optimization study

We present a program used to calculate intricate three-particle integrals for variational calculations of solutions to the leptonic Schrödinger equation with two nuclear centres in which inter-leptonic distances (electron–electron and positron–electron) are included directly in the trial functions. The program has been used so far in calculations of He – H ¯ interactions and positron H 2 scattering, however the precisely defined integrals are applicable to other situations. We include a summary discussion of how the program has been optimized from a ‘legacy’-type code to a more modern high-performance code with a performance improvement factor of up to 1000. Program summary Program title: tripleint.cc Catalogue identifier: AEEV_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEEV_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 12 829 No. of bytes in distributed program, including test data, etc.: 91 798 Distribution format: tar.gz Programming language: Fortran 95 (fixed format) Computer: Modern PC (tested on AMD processor) [1], IBM Power5 [2] Cray XT4 [3], similar Operating system: Red Hat Linux [1], IBM AIX [2], UNICOS [3] Has the code been vectorized or parallelized?: Serial (multi-core shared memory may be needed for some large jobs) RAM: Dependent on parameter sizes and option to use intermediate I/O. Estimates for practical use: 0.5–2 GBytes (with intermediate I/O); 1–4 GBytes (all-memory: the preferred option). Classification: 2.4, 2.6, 2.7, 2.9, 16.5, 16.10, 20 Nature of problem: The ‘tripleint.cc’ code evaluates three-particle integrals needed in certain variational (in particular: Rayleigh–Ritz and generalized-Kohn) matrix elements for solution of the Schrödinger equation with two fixed centres (the solutions may then be used in subsequent dynamic nuclear calculations). Specifically the integrals are defined by Eq. (16) in the main text and contain terms proportional to r i k × r j k / r i j , i ≠ j , i ≠ k , j ≠ k , with r i j the distance between leptons i and j. The article also briefly describes the performance optimizations used to increase the speed of evaluation of the integrals enough to allow detailed testing and mapping of the effect of varying non-linear parameters in the variational trial functions. Solution method: Each integral is solved using prolate spheroidal coordinates and series expansions (with cut-offs) of the many-lepton expressions. 1-d integrals and sub-integrals are solved analytically by various means (the program automatically chooses the most accurate of the available methods for each set of parameters and function arguments), while two of the three integrations over the prolate spheroidal coordinates ‘ λ’ are carried out numerically. Many similar integrals with identical non-linear variational parameters may be calculated with one call of the code. Restrictions: There are limits to the number of points for the numerical integrations, to the cut-off variable itaumax for the many-lepton series expansions, and to the maximum powers of Slater-like input functions. For runs near the limit of the cut-off variable and with certain small-magnitude values of variational non-linear parameters, the code can require large amounts of memory (an option using some intermediate I/O is included to offset this). Unusual features: In addition to the program, we also present a summary description of the techniques and ideology used to optimize the code, together with accuracy tests and indications of performance improvement. Running time: The test runs take 1–15 minutes on HPCx [2] as indicated in Section 5 of the main text. A practical run with 729 integrals, 40 quadrature points per dimension and itaumax = 8 took 150 minutes on a PC (e.g., [1]): a similar run with ‘medium’ accuracy, e.g. for parameter optimization (see Section 2 of the main text), with 30 points per dimension and itaumax = 6 took 35 minutes.

Improved (1s)2 variational model for helium

Abstract The ground-state energy of neutral helium is estimated variationally with a trial wavefunction of the form ϕ≈e −γ(rA/a o)ne−γ(rB/a o)n. This model represents a modification of traditional textbook examinations of this problem via inclusion of the power “n” as a second nonlinear variational parameter in addition to the usual effective nuclear charge γ and leads to an upper-limit on the ground state energy of −2.86107 Eh (Eh=1 hartree) in comparison with the traditional (n=1) result of −2.84766 Eh. This result represents a reduction of the percentage overestimate from the true ground-state energy (−2.90373 Eh) of from 1.93 to 1.47. In comparison with the maximum accuracy obtainable from an uncorrelated trial wavefunction, −2.86168 Eh, the present trial wavefunction reduces the percentage overestimate from 0.49 (n=1) to 0.021. The optimum values of (n, γ) are determined to be ≈(0.897, 1.825).

The Condition of Invariance of the Energy of an Atomic-Molecular System with Respect to Rotations of the Jacobi Coordinates

An additional condition for the wave functions of the ground and excited states of atomic-molecular systems, which follows from the energy invariance with respect to rotations of the Jacobi coordinates, is studied. This condition makes a substantial contribution to the conventional criterion of the quality of wave functions, based on the variational principle for the average energy value. The explicit form of this condition is found for an arbitrary interaction potential dependent on the particle-particle distances and the requirements for realization of this condition are ascertained. Variational calculations of 3He2+ μ−e− and 4He2+ μ−e− mesoatoms in the basis of correlated exponential functions dependent on the particle-particle distances are performed with a detailed optimization of all nonlinear parameters for each individual energy level. A high sensitivity of the condition under study to the quality of approximate wave functions and the efficiency of the detailed optimization of the nonlinear variational parameters in calculation of each individual energy level are demonstrated.

On the optimization of Gaussian basis sets

A new procedure for the optimization of the exponents, αj, of Gaussian basis functions, Ylm(ϑ,φ)rle−αjr2, is proposed and evaluated. The direct optimization of the exponents is hindered by the very strong coupling between these nonlinear variational parameters. However, expansion of the logarithms of the exponents in the orthonormal Legendre polynomials, Pk, of the index, j: ln αj=∑k=0kmaxAkPk((2j−2)/(Nprim−1)−1), yields a new set of well-conditioned parameters, Ak, and a complete sequence of well-conditioned exponent optimizations proceeding from the even-tempered basis set (kmax=1) to a fully optimized basis set (kmax=Nprim−1). The error relative to the exact numerical self-consistent field limit for a six-term expansion is consistently no more than 25% larger than the error for the completely optimized basis set. Thus, there is no need to optimize more than six well-conditioned variational parameters, even for the largest sets of Gaussian primitives.

Simultaneous optimization of molecular geometry and the wave function in a basis of Singer's n-electron explicitly correlated Gaussians

Analytical gradients of the Born–Oppenheimer variational energy of a molecular system with respect to nuclear coordinates are derived and implemented to optimize the molecular geometry in a basis of Singer's explicitly correlated Gaussian functions. The 3 N−6 geometry variables are optimized simultaneously with the linear and non-linear variational parameters of the wave function. The method can be used for global minimization, transition structure prediction, and following reaction paths in problems that require very accurate wave functions. Test results on hydrogen clusters with two and three atoms are presented as an illustration of the method.

Non-Born–Oppenheimer calculations on the LiH molecule with explicitly correlated Gaussian functions

We report the first ever non-Born–Oppenheimer variational calculations on the ground state of a four electron molecular system. The basis set used in the calculations consists of explicitly correlated Gaussians multiplied by powers of the internuclear distance. To accelerate the optimization of the many nonlinear variational parameters involved in the variational wave function, we performed the calculations on a cluster of Linux workstations using MPI and a parallel implementation of the formulas. Results for the nonadiabatic ground state energy of LiH, as well as expectation values for the kinetic and potential energies, the internuclear and square of the internuclear distance, the virial coefficient, and the square of the energy gradient norm are reported.

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

AbstractAnalytical 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

A variational calculation exhibiting both minima and maxima

A simple example of a variational calculation for the quantum mechanical energies of a system is presented. The trial function employed has only one (nonlinear) variational parameter. We point out a rather unexpected result for the energy extrema associated with this parameter.

Variational principle for critical parameters of quantum systems

The variational principle for eigenvalue problems with a nonidentity weight operator is used to establish upper or lower bounds on critical parameters of quantum systems. Three problems from atomic physics are considered as examples. Critical screening parameters for the exponentially screened Coulomb potential are found using a trial function with one nonlinear variational parameter. The critical charge for the helium isoelectronic series is found using a Hylleraas-type trial function. Finally, critical charges for the same system subjected to a magnetic field are found using a product of two hydrogen-like basis sets.

Monte Carlo optimisation of correlated wave functions for normal two-electron systems

A simple, new type of correlated wave function is proposed for the studies of normal two-electron atomic systems: ψ( r 1 , r 2 ) = Σc m Φ m ( r 1 , r 2 ) with Φ m ( r 1 , r 2 ) = exp [−α(r 1 + r 2 )] (br 12 + a) m , where α, a , b are non-linear variational parameters. A notable feature of this basis function is that only three terms are required within the framework of the Raleigh-Ritz variational principle to obtain fairly accurate energy eigenvalues and satisfactory cusp conditions. The non-linear variational parameters are optimised by using the Monte Carlo technique.

Monte-Carlo optimization of correlated helium wave function

A new type of correlated wave function for the normal helium is proposed in the form:ψ(r1, r2)=Σc mφm withφ m=exp]−α(r 1+r 2)]/(br 12+a) m wherea, b, α are non-linear variational parameters. Optimizing these parameters by the Monte-Carlo technique, an energy eigen-value of −2.903645(a.u.) is obtained withonly three terms in the basis expansion alongwith satisfactory cusp condition, compared to the essentially exact non-relativistic value of −2.903724 as given by Freundet al (Phys. Rev. A29, 980 (1984)).

Accurate relativistic energies of one- and two-electron systems using Gaussian wave functions

Gaussian wave functions with optimized nonlinear variational parameters are applied to evaluate clamped-nucleus nonrelativistic energies as well as the lowest order relativistic corrections for the ground states of the hydrogen atom, hydrogen molecular ion H+2, helium atom, and hydrogen molecule. The two-electron functions used depend explicitly on the interelectronic distance r12 but do not describe the cusp properly. Despite this, for H+2 and H2 the results are more accurate than ever reported, and for He they are inferior only to the best calculations employing Hylleraas-type expansions. It is demonstrated that, contrary to a common opinion, Gaussian wave functions are very well suited for high-accuracy relativistic computations even in the Breit–Pauli approximation, provided that the nonlinear parameters are optimized with respect to the nonrelativistic energy.

Some calculations on the ground and lowest-triplet state of helium in the fixed-nucleus approximation.

The series solution method developed by Pekeris [Phys. Rev. 112, 1649 (1958); 115, 1216 (1959)] for the Schroedinger equation for two-electron atoms, as generalized by Frost [ital et] [ital al]. [J. Chem. Phys. 41, 482 (1964)] to handle any three particles with a Coulomb interaction, has been used. The wave function is expanded in triple orthogonal set in three [ital perimetric] coordinates. From the Schroedinger equation an explicit recursion relation for the coefficients in the expansion is obtained, and the vanishing of the determinant of these coefficients provides the condition for the energy eigenvalues and for the eigenvectors. The Schroedinger equation is solved and the matrix elements are produced algebraically by using the computer algebra system MAPLE. The substitutions for a particular atom and diagonalization were performed by a program written in the C language. Since the determinant is sparse, it is possible to go to the order of 1078 as Pekeris did without using excessive memory or computer CPU time. By using a nonlinear variational parameter in the expression used to remove the energy, nonrelativistic energies, within the fixed-nucleus approximation, have been obtained. For the ground-state singlet 1 [sup 1][ital S] state, this is of the accuracy claimed bymore » Frankowski and Pekeris [Phys. Rev. 146, 46 (1966); 150, 366(E) (1966)] using logarithmic terms for [ital Z] from 1 to 10, and for the triplet 2 [sup 3][ital S] state, energies have been obtained to 12 decimal places of accuracy, which, with the exception of [ital Z]=2, are lower than any previously published, for all [ital Z] from 3 to 10.« less

Atomic orbital basis set optimization for ab initio calculations of molecules with hydrogen atoms in strong magnetic fields

We present the results of the optimization of a recently established atomic orbital basis set for ab initio calculations of molecules in strong magnetic fields. The optimization is performed for the ground and many low-lying excited states of the hydrogen atom in the presence of a magnetic field of arbitrary strength. The behavior of the nonlinear variational parameters of the atomic orbitals is discussed as a function of the field strength. In order to demonstrate the suitability of our atomic orbital basis set for the calculation of molecular spectra and wave functions in a uniform magnetic field we present, as first results of a molecular calculation, electronic potential energy curves for the H+2 ion in a magnetic field.

Quadrupole and octupole Cauchy moments of the atoms through argon

AbstractQuadrupole and octupole Cauchy moments of the atoms through argon are calculated using the hydrodynamic formulation of time‐dependent Kohn–Sham theory. The exchange‐correlation energy density functional is approximated by a gradient expansion for atoms that has an explicit dependence upon the number of electrons. The first‐order corrections to the Kohn–Sham amplitudes and phases are found by seeking variational solutions of the derived sequential set of functionals. The trial functions employed contain both linear and nonlinear variational parameters and are thus flexible enough to provide rapid convergence to the multipole polarizabilities. The resulting Cauchy moments provide information that allows the calculation of various properties that result from the linear interaction of atoms with a time‐varying electric field. © 1994 John Wiley & Sons, Inc.

Dipole Cauchy moments of the atoms H through Ar

Dipole Cauchy moments of the atoms through Ar are calculated using the hydrodynamic formulation of time-dependent Kohn-Sham theory. The exchange-correlation energy density functional is approximated by the gradient expansion for atoms. Using variational trial functions that contain both linear and nonlinear variational parameters, we are able to reproduce (to three or four significant figures) the static dipole polarizabilities obtained by explicitly solving the relevant differential equations. The resulting dipole Cauchy moments provide a convenient starting point for calculating other properties which result from the linear interaction of atoms with a time varying electric field.