Compact Wave Functions Research Articles

Application of variational Monte Carlo method to the confined helium atom

A new application of variational Monte Carlo method is presented to study the helium atom under the compression effect of a spherical box with radius (rc). The ground-state energies of the helium atom were calculated for different values of rc. Our calculations were extended to include Li + and Be 2+ ions. The calculations were based on the use of a compact accurate trial wave function with five variational parameters. To optimize variational parameters, we used the steepest descent method. The obtained results are in good agreement with previous results.

Generalized variational principle for excited states using nodes of trial functions

The familiar variational principle provides an upper bound to the ground-state energy of a given Hamiltonian. This allows one to optimize a trial wave function by minimizing the expectation value of the energy. This approach is also trivially generalized to excited states, so that given a trial wave function of a certain symmetry, one can compute an upper bound to the lowest-energy level of that symmetry. In order to generalize further and build an upper bound of an arbitrary excited state of the desired symmetry, a linear combination of basis functions is generally used to generate an orthogonal set of trial functions, all bounding their respective states. However, sometimes a compact wave-function form is sought, and a basis-set expansion is not desirable or possible. Here we present an alternative generalization of the variational principle to excited states that does not require explicit orthogonalization to lower-energy states. It is valid for one-dimensional systems and, with additional information, to at least some n-dimensional systems. This generalized variational principle exploits information about the nodal structure of the trial wave function, giving an upper bound to the exact energy without the need to build a linear combination of basis functions. To illustrate the theorem we apply it to a nontrivial example: the 1s2s (1)S excited state of the helium atom.

Quantum Monte Carlo with Jastrow-valence-bond wave functions

We consider the use in quantum Monte Carlo calculations of two types of valence bond wave functions based on strictly localized active orbitals, namely valence bond self-consistent-field and breathing-orbital valence bond wave functions. Complemented by a Jastrow factor, these Jastrow-valence-bond wave functions are tested by computing the equilibrium well depths of the four diatomic molecules C(2), N(2), O(2), and F(2) in both variational Monte Carlo and diffusion Monte Carlo. We show that it is possible to design compact wave functions based on chemical grounds that are capable of describing both static and dynamic electron correlations. These wave functions can be systematically improved by inclusion of valence bond structures corresponding to additional bonding patterns.

Compact and accurate variational wave functions of three-electron atomic systems constructed from semi-exponential radial basis functions

The semi-exponential basis set of radial functions (A.M. Frolov, Physics Letters A {\bf 374}, 2361 (2010)) is used for variational computations of bound states in three-electron atomic systems. It appears that semi-exponential basis set has a substantially greater potential for accurate variational computations of bound states in three-electron atomic systems than it was originally anticipated. In particular, the 40-term Larson's wave function improved with the use of semi-exponential radial basis functions now produces the total energy \linebreak -7.47805413551 $a.u.$ for the ground $1^2S-$state in the ${}^{\infty}$Li atom (only one spin function $\chi_1 = \alpha \beta \alpha - \beta \alpha \alpha$ was used in these calculations). This variational energy is very close to the exact ground state energy of the ${}^{\infty}$Li atom and it substantially lower than the total energy obtained with the original Larson's 40-term wave function (-7.477944869 $a.u.$).

Multilayer multiconfiguration time-dependent Hartree method: Implementation and applications to a Henon–Heiles Hamiltonian and to pyrazine

The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method is discussed and a fully general implementation for any number of layers based on the recursive ML-MCTDH algorithm given by Manthe [J. Chem. Phys. 128, 164116 (2008)] is presented. The method is applied first to a generalized Henon-Heiles (HH) hamiltonian. For 6D HH the overhead of ML-MCTDH makes the method slower than MCTDH, but for 18D HH ML-MCTDH starts to be competitive. We report as well 1458D simulations of the HH hamiltonian using a seven-layer scheme. The photoabsorption spectrum of pyrazine computed with the 24D hamiltonian of Raab et al. [J. Chem. Phys. 110, 936 (1999)] provides a realistic molecular test case for the method. Quick and small ML-MCTDH calculations needing a fraction of the time and resources of reference MCTDH calculations provide already spectra with all the correct features. Accepting slightly larger deviations, the calculation can be accelerated to take only 7 min. When pushing the method toward convergence, results of similar quality than the best available MCTDH benchmark, which is based on a wavepacket with 4.6×10(7)time-dependent coefficients, are obtained with a much more compact wavefunction consisting of only 4.5×10(5) coefficients and requiring a shorter computation time.

Compact variational wave functions for bound states in three-electron atomic systems

The variational procedure to construct compact and accurate wave functions for three-electron atoms and ions is developed. The procedure is based on the use of six-dimensional Gaussoids written in the relative four-body coordinates r 12, r 13, r 23, r 14, r 24, and r 34. The nonlinear parameters in each basis function have been carefully optimized. Using these variational wave functions, we have determined the energies and other bound state properties for the ground 12 S-states in a number of three-electron atoms and ions. The three-electron atomic systems considered in this work include the neutral Li atom and nine positively charged lithiumlike ions: Be+, B2+, C3+, ..., Na8+, and Mg9+. Our variational wave functions are used to determine the hyperfine structure splitting and field shifts for some lithium-like ions. The explicit formulas of the Q −1 expansion are derived for the total energies of these three-electron systems.

Compact wave functions for four-electron atomic systems

A procedure is developed to construct improved wave functions for the bound states in various four-electron atoms and ions. The bound state wave functions are improved in the sense that our trial wave functions are compact and accurate. We employ two independent four-electron spin functions ${\ensuremath{\chi}}_{1}=\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}+\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}$ and ${\ensuremath{\chi}}_{2}=2\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}+2\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}$ in each trial wave function. Optimization of the nonlinear parameters is performed for each radial function. Our procedure is applied to obtain the total energies of the ground $^{1}S^{e}$ states in the following four-electron berylliumlike systems: ${\mathrm{Li}}^{\ensuremath{-}}$, Be, ${\mathrm{B}}^{+}$, ${\mathrm{C}}^{2+}$, ${\mathrm{N}}^{3+}$, ${\mathrm{O}}^{4+}$, ${\mathrm{F}}^{5+}$, ${\mathrm{Ne}}^{6+}$, ${\mathrm{Na}}^{7+}$, and ${\mathrm{Mg}}^{8+}$. All our trial wave functions used in this study are very compact. In fact, each of these functions contains only 300 basis (radial) functions which are constructed from ten-dimensional gaussoids of the five-body relative coordinates ${r}_{12}$, ${r}_{13}$, ${r}_{14}$, ${r}_{15}$, ${r}_{23}$, ${r}_{24}$, ${r}_{25}$, ${r}_{34}$, ${r}_{35}$, and ${r}_{45}$.

A compact boundary-condition- determined wavefunction for two-electron atomic systems

Highly compact wavefunctions with a clear physical meaning for the He atom and He-like isoelectronic ions for Z = 1–10 are written as a symmetrized product of exp[(ar + br2)/(1 + r)] electron–nucleus terms and an electron–electron Jastrow factor to satisfy the correct asymptotic behaviour both at short and long interparticle distances. Some parameters are chosen to satisfy exactly the cusp conditions, while the others are optimized by variational Monte Carlo calculations. The wavefunction energy is within 2 millihartrees from the non-relativistic limit in the entire Z-range, improving previously published work on similar compact wavefunctions. We tested the validity of the ‘coalescence wavefunction’ approximation. The Z-dependence of the optimized parameters allows us to write a general form of the wavefunction, using Z as an explicit parameter and four parameters independent of Z. We checked the validity of this wavefunction on the case Z = 30.

Study of correlation effects during reemission of ultrashort electromagnetic pulses by the helium atom

The calculation of the energy of the ground state of atoms is the criterion for the correctness of compact analytical atomic wave functions. However, such a test is in fact static. We studied the degree of taking into account electron correlations in various simple analytical wave functions in the dynamic process of reemission by the helium atom of ultrashort pulses of an electromagnetic field. Direct guidelines are given on the use of particular analytical wave functions, which can be helpful in simple calculations and estimations of dynamic processes.

K-shell andL-shell(e,3e)double ionization of beryllium by fast electron impact

We report fully differential cross sections for simultaneous double ionization of atomic beryllium by fast electron impact. We analyze separately the two ionization channels that leave the dication in either the $2{s}^{2}$ autoionizing excited state or its $1{s}^{2}$ bound ground state. For the double continuum, owing to the two slow emitted electrons, we employ the three-pairwise-Coulomb-interaction model (3C) along with the first Born approximation. Aiming to probe the hitherto largely unexplored role of the electron correlation in $(e,3e)$, we determine and check fully correlated, compact analytical wave functions, satisfying all two-particle Kato cusp conditions for the four-electron Be initial state and the two-electron ${\text{Be}}^{2+}$ final states, as well as other, much or little or noncorrelated, functions, issuing from either analytic global optimization or Hartree-Fock theory. We point out similarities or contrasts with trends observed in the double photoionization, allowing for parallels between $(e,3e)$ and $(\ensuremath{\gamma},2e)$. Various numerical examples serve to navigate possible future experiments in the nonrelativistic regime of the ionization problem.

A variationally stable compact Hartree–Fock-style wavefunction for a non-degenerate first excited state

A compact approximate wavefunction for a non-degenerate first excited state constructed as a Slater determinant from which the ground state Hartree–Fock (HF) wavefunction has been projected out explicitly is introduced, together with all expressions required for its calculation through gradient-based energy optimization schemes. The explicit projection makes this wavefunction suitable for the description of a non-degenerate first excited state which is of the same symmetry as the HF ground state without any danger of a variational collapse to the lower energy ground state. The equations allowing the calculation of a spin-unrestricted version of the proposed wavefunction have been implemented in code. The results of excited state calculations on HeH+, H2, C2, N2, H2O and H2O2 show that this excited state wavefunction which, effectively, is comprised of just two Slater determinants, only one of which is optimized variationally, is capable of producing results which compare favourably to those coming from standard approaches such as single-excitation configuration interaction (CIS), time-dependent HF (TDHF), symmetry-adapted-cluster CI (SAC-CI) and equations-of-motion coupled-cluster with singles and doubles (EOM-CCSD).

Requirements for the generalization of the ab initio two-state model for external electric field induced electron transfer at electrodes

Abstract This work reports a careful systematic analysis of the validity of the ab initio two-state model for electron transfer between halides and various transition metal single crystal electrodes induced by an external electric field. The surfaces are represented by cluster models and the electron transfer is modeled by compact ab initio wave functions, which are a linear combination of donor–acceptor valence bond structures (DA and D + A − ) written in an appropriate orbital basis. It is found that the two-state model is able to successfully describe the ET for all those adsorbate/surface systems that feature a sufficiently large vertical equilibrium distance (orbitals well localized on the fragments and VB structures properly defined), and that exhibit at least a partially ionic character. The former condition occurs always when the solvent is explicitly taken into account, whereas the latter is fulfilled when not complete charge transfer occurs upon chemisorption.

The strangeness form factors of the proton

The present empirical information on the strangeness form factors indicates that the corresponding uudss¯ component in the proton is such that the uuds subsystem has the flavor spin symmetry [4]FS[22]F[22]S and mixed orbital symmetry [31]X. This uudss¯ configuration leads to the empirical signs of all the form factors GEs,GMs and GAs. An analysis with simple quark model wave functions for the preferred configuration shows that the qualitative features of the empirical strangeness form factors may be described with a ∼15% admixture of uudss¯ with a compact wave function in the proton. Transition matrix elements between the uud and uudss¯ components give significant contributions.

Lewis-Based Valence Bond Scheme: Application to the Allyl Cation

A method for expressing the wave function in terms of Lewis structures is proposed and tested on the allyl cation. This computational scheme is called valence bond BOND (VBB). The compact VBB wave function gives consistent results with the breathing orbital valence bond method (BOVB) for the resonance energy of the allyl cation (54 and 55 kcal/mol for VBB and BOVB, respectively). The optimization of the sigma orbitals, in such a way they adapt to each resonance structure, makes use of the breathing orbital effect. It is shown that this "breathing" of the sigma frame is more efficient in the resonant hybrid than in the localized state, so that a resonance energy of 63 kcal/mol is obtained at this level of computation.

On the orbital dependence of compact, weight-selected configuration interaction and coupled-cluster wave functions

We recently reported that very compact coupled-cluster wave functions may be generated by selecting the most important configurations, by weight, from the full coupled-cluster wave function. Here, we consider how the choice of orbitals may affect these wave functions in the case of the symmetric dissociation of H2O. We employ unrestricted Hartree–Fock and complete-active-space self-consistent-field orbitals, as well as natural orbitals derived from a coupled-cluster singles and doubles wave function. For a given accuracy, some choices of orbitals can reduce the size of configuration interaction wave functions, but they have little effect on the weight-selected coupled-cluster wave functions.

Highly Compact Wave Functions for He-Like Systems

Wave functions which are compact, but still quite accurate, are extremely valuable as tools for gaining understanding of quantum systems. This paper investigates the use for that purpose of functions that depend exponentially on all the interparticle distances of a few-body system, illustrated by a study of the ground electronic states of the He isoelectronic series (Z from 1 to 10). Using as few as 4 exponential basis functions, it is found that nonrelativistic energies are reproduced to within 38 microhartrees of the exact values, an error far less than for previously reported compact wave functions. Other properties are also well-represented.

Correlation energy extrapolation by intrinsic scaling. III. Compact wave functions

The information gained in the context of extrapolating the correlation energy by intrinsic scaling is used to shorten the full configurational expansions of electronic wave function without compromising their chemical accuracy. The truncations are accomplished by judiciously limiting the participation of the ranges of predetermined approximate sets of natural orbitals in the various excitation categories.

Correlation energy extrapolation by intrinsic scaling. II. The water and the nitrogen molecule

The extrapolation method for determining benchmark quality full configuration-interaction energies described in preceding paper [L. Bytautas and K. Ruedenberg, J. Chem. Phys. 121, 10905 (2004)] is applied to the molecules H(2)O and N(2). As in the neon atom case, discussed in preceding paper [L. Bytautas and K. Ruedenberg, J. Chem. Phys. 121, 10905 (2004)] remarkably accurate scaling relations are found to exist between the correlation energy contributions from various excitation levels of the configuration-interaction approach, considered as functions of the size of the correlating orbital space. The method for extrapolating a sequence of smaller configuration-interaction calculations to the full configuration-interaction energy and for constructing compact accurate configuration-interaction wave functions is also found to be effective for these molecules. The results are compared with accurate ab initio methods, such as many-body perturbation theory, coupled-cluster theory, as well as with variational calculations wherever possible.

Baryon form factors of relativistic constituent-quark models

The electromagnetic and axial form factors of the nucleon and its lowest positive parity excitations, the Delta(1232) and the N*(1440), are calculated with constituent-quark models that are specified by simple algebraic representations of the mass-operator eigenstates. Poincar\'e covariant current operators are generated by the dynamics from single-quark currents that are covariant under a kinematic subgroup. The dependence of the calculated form factors on the choice of kinematics and on the gross features of the wave functions is illustrated for instant-form, point-form, and front-form kinematics. A simple algebraic form of the orbital ground state wave function, which depends on two parameters, allows a fair description of all the form factors over the empirically accessible range, although with widely different choices of the parameters, which determine the range and shape of the orbital wave function. The neutron electric form factor requires additional features, for instance the presence of mixed symmetry S-state component with 1 -- 2 % probability in the ground state wave function. Instant and front form kinematics demand a spatially extended wave function, whereas in point form kinematics the form factors may be described with a quite compact wave function.

Scaling of hadronic form factors in point form kinematics

The general features of baryon form factors calculated with point form kinematics are derived. With point form kinematics and spectator currents hadronic form factors are functions of η:= 1 4 (v out −v in ) 2 and, over a range of η values, are insensitive to unitary scale transformations of the model wave functions when the extent of the wave function is small compared to the scale defined by the constituent mass, 〈 r 2〉⪡1/ m 2. The form factors are sensitive to the shape of such compact wave functions. Simple 3-quark proton wave functions are employed to illustrate these features. Rational and algebraic model wave functions lead to a reasonable representation of the empirical form factors, while Gaussian wave functions fail. In point form kinematics the form factors have nontrivial behavior in the limit 〈 r 2〉 m 2=0 with power law behavior for large values of η.