Full Configuration Interaction Wave Research Articles

Origin of Hund’s multiplicity rule in singly excited helium: Existence of a conjugate Fermi hole in the lower spin state

The origin of Hund's multiplicity rule in the low-lying excited states of the helium atom has been studied by considering the two-dimensional helium atom. The internal part of the full configuration interaction wave functions for the (2s) and (2p) singlet-triplet pairs of states has been extracted and visualized in the three-dimensional internal space (r{sub 1},r{sub 2}, {phi}{sub -}). The internal wave function of the singlet states without electron repulsion has a significant probability around the origin of the internal space while the corresponding probability of the triplet wave function is negligible in this region due to the presence of a Fermi hole. The electron-electron repulsion potential has been visualized also in the internal space. It manifests itself by three striking poles penetrating exactly into the spatial region defined by the Fermi hole. Because of the existence of these strong potential poles in the vicinity of the Fermi hole a major part of the singlet probability migrates out of this region. In contrast, the corresponding triplet wave function is less affected by these poles due to the presence of the Fermi hole. The singlet probability is shown to migrate from its original region close to the origin to a region farmore » away where either r{sub 1} or r{sub 2} are large. This results in a more diffuse electron density distribution and a smaller electron repulsion energy of the singlet state than of the corresponding triplet state. The mechanism of the evolution of the singlet probability toward the region of large r{sub i} (i =1, 2) in the presence of the electron repulsion potential has been rationalized on the basis of a new concept called conjugate Fermi hole.« less

Origin of Hund’s multiplicity rule in quasi-two-dimensional two-electron quantum dots

The origin of Hund's multiplicity rules has been studied for a system of two electrons confined by a quasi-two-dimensional harmonic-oscillator potential by relying on a full configuration interaction wave function and Cartesian anisotropic Gaussian basis sets. In terms of appropriate normal-mode coordinates the wave function factors into a product of the center-of-mass and the internal components. The ${}^{1}{\ensuremath{\Pi}}_{u}$ singlet state and the ${}^{3}{\ensuremath{\Pi}}_{u}$ triplet state represent the energetically lowest pair of states to which Hund's multiplicity rule applies. They are shown to involve excitations into different degrees of freedom, namely, into the center-of-mass angular mode and the internal angular mode for the singlet and triplet states, respectively. The presence of an angular nodal line in the internal space allows then the triplet state to avoid the singularity in the electron-electron interaction potential, leading to the energy lowering of the triplet state relative to its counterpart singlet state.

Full configuration interaction wave function as a formal solution to the optimized effective potential and Kohn–Sham models in finite basis sets

Using finite basis sets, it is shown how to construct a local Hamiltonian, such that one of its infinitely many degenerate eigenfunctions is the ground-state full configuration interaction (FCI) wave function in that basis set. Formally, the local potential of this Hamiltonian is the optimized effective potential and the exact Kohn–Sham potential at the same time, because the FCI wave function yields the exact ground-state density and energy. It is not the aim of this paper to provide a new algorithm for obtaining FCI wave functions. A new potential is the goal.

Direct calculation of density matrices: Natural orbitals and occupation numbers of model conjugated molecules

The occupation numbers and natural orbitals of several model π-electron molecules are presented and analyzed, which are directly calculated by employing a theory developed earlier by one of the authors. Computational details are discussed briefly, and the errors which arise in our calculations are estimated through a comparison with the density matrix of a full configuration interaction wave function for the pyridine molecule.

Magnitude and significance of the higher‐order reduced density matrix cumulants

AbstractUsing full configuration interaction wave functions for Be and LiH, in both minimal and extended basis sets, we examine the absolute magnitude and energetic significance of various contributions to the three‐electron reduced density matrix (3‐RDM) and its connected (size‐consistent) component, the 3‐RDM cumulant (3‐RDMC). Minimal basis sets are shown to suppress the magnitude of the 3‐RDMC in an artificial manner, whereas in extended basis sets, 3‐RDMC matrix elements are often comparable in magnitude to the corresponding 3‐RDM elements, even in cases where this result is not required by spin angular momentum coupling. Formal considerations suggest that these observations should generalize to higher‐order p‐RDMs and p‐RDMCs (p > 3). This result is discussed within the context of electronic structure methods based on the contracted Schrödinger equation (CSE), as solution of the CSE relies on 3‐ and 4‐RDM “reconstruction functionals” that neglect the 3‐RDMC, the 4‐RDMC, or both. Although the 3‐RDMC is responsible for at most 0.2% of the total electronic energy in Be and LiH, it accounts for up to 70% of the correlation energy, raising questions regarding whether (and how) the CSE can offer a useful computational methodology. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Important configurations in configuration interaction and coupled-cluster wave functions

Using a new general-order determinant-based program, we construct compact configuration interaction and coupled-cluster wave functions by selecting the most important configurations, by weight, from a full configuration interaction or full coupled-cluster wave function. Our results show that for the symmetric dissociation of water, chemical accuracy can be achieved across the surface with ∼2% of the full coupled-cluster expansion compared to ∼10% of the full configuration interaction expansion.

Full configuration interaction calculation of Be3.

The full configuration interaction (FCI) study of the ground state of the neutral beryllium trimer has been performed using an atomic natural orbitals [3s2p1d] basis set. Both triangular and linear structures have been considered for the Be(3) cluster. The optimal geometry for the equilateral triangle has been calculated. The potential energy cut sections along the normal a(1)(') mode and one of the components of the e(') mode have then been studied. The FCI symmetric atomization potential of the linear cluster is also reported. It shows a secondary van der Waals minimum at a long bond distance. All singular points in the potential energy curves are characterized. Other properties, like dissociation energies D(e) and vibrational frequencies, have been estimated from a fourth-order fitting of a large range of points around the minima. The calculated FCI wave number values for the nu(1) and nu(2) normal modes are (467.33+/-0.43) cm(-1) and (390.77+/-0.56) cm(-1).

The general-model-space state-universal coupled-cluster method exemplified by the LiH molecule

The salient features of the recently introduced general-model-space (GMS) state-universal (SU) coupled-cluster (CC) method are illustrated on the case of the LiH molecule. Describing the breaking of the Li–H bond by relying on an open-shell-type GMS reveals the importance of the connectivity conditions (C conditions), which represent a crucial new ingredient of the GMS SU CC theory. Only when we properly account for these C conditions can we uniquely represent the full configuration interaction (FCI) wave functions in terms of the multireference SU exponential cluster ansatz and recover the FCI energies via the GMS SU CC method, assuming that all the relevant clusters at a given level of the theory are considered. Drawing on various GMSs, we compute the potential energy curves for three Σ+1, two Σ+3, three Π,1 and three Π3 states, using the GMS SU CC method truncated at the singly- and doubly-excited level (GMS SU CCSD), as well as the externally corrected (N,M)-CCSD method that exploits the NR-CISD wave functions as the external source of higher-than-pair clusters in the MR SU CCSD method. In all cases we obtain excellent results: For Σ+ states, the maximum difference between the FCI and various SU CCSD energies is about 0.5 millihartree. These errors are further reduced when we employ the (N,M)-CCSD methods. For the Π states, the deviations of the SU CCSD energies relative to FCI amount to at most a few hundreds of a millihartree. We also report on the size-extensivity tests and the exactness of the formalism for two-electron systems.

General-model-space state-universal coupled-cluster theory: Connectivity conditions and explicit equations

We present a new version of the state-universal (SU), multireference, coupled-cluster (CC) theory that is capable of handling completely general, incomplete model spaces. This is achieved by exploiting the concept of “locality” for the active molecular spin orbitals and by introducing the constraining conditions (C conditions) on cluster amplitudes that are associated with the internal excitations transforming one reference configuration into another one. These C conditions make it possible to represent the exact (i.e., full configuration interaction) wave function via the SU CC cluster ansatz based on an arbitrary model space. The C conditions are then taken into account together with the standard SU CC equations for the external amplitudes, thus enabling us to reach the exact result in the limit, while preserving the connectivity property and thus the size extensivity. We also present compact expressions for the matrix elements of the effective Hamiltonian as well as the explicit expressions for the most important coupling coefficients that are required at the single and double excitation level. All other expressions are the same as in the single reference CC formalism.

Fine and hyperfine structure in three low-lying 3Σ+ states of molecular hydrogen

The fine structure constant (electron spin-spin coupling) and the hyperfine structure parameters (electron-nuclear spin coupling, including spin-rotation and electron-nuclear quadrupole coupling) in the low-lying triplet states b3Σ+ u, a3Σ+ g and e3Σ+ u of molecular hydrogen and deuterium are calculated using a recently developed technique with full configuration interaction and multiconfiguration self-consistent field wave functions. The second-order spin-orbit coupling contribution to the 3Σ+ states splitting is negligible, and the calculations therefore provide a good estimate of the zero-field splitting based only on the electron spin-spin coupling values. For the bound a3Σ+ g state a negligible zero-field splitting is found, in qualitative agreement with the e-a spectrum. The zero-field splitting parameter is considerable for the repulsive b3Σ+ u state (≃1 cm−1) and of intermediate size for the bound e3Σ+ u state. The isotropic hyperfine coupling constant is very large not only for the valence b3Σ+ u state (1580 MHz) but also for the Rydberg a and e triplet states (≃1400 MHz). The quadrupole coupling constants for the deuterium isotopes are negligible (0.04–0.07 MHz) for all studied triplet states. The electric dipole activity of the spin sublevels in the triplet-singlet transitions to the ground state is estimated by means of the quadratic response technique.

Perturbative corrections to coupled-cluster and equation-of-motion coupled-cluster energies: A determinantal analysis

We develop a combined coupled-cluster (CC) or equation-of-motion coupled-cluster (EOM-CC) theory and Rayleigh–Schrödinger perturbation theory on the basis of a perturbation expansion of the similarity-transformed Hamiltonian H̄=exp(−T)H exp(T). This theory generates a series of perturbative corrections to any of the complete CC or EOM-CC models and hence a hierarchy of the methods designated by CC(m)PT(n) or EOM-CC(m)PT(n). These methods systematically approach full configuration interaction (FCI) as the perturbation order (n) increases and/or as the cluster and linear excitation operators become closer to complete (m increases), while maintaining the orbital-invariance property and size extensivity of CC at any perturbation order, but not the size intensivity of EOM-CC. We implement the entire hierarchy of CC(m)PT(n) and EOM-CC(m)PT(n) into a determinantal program capable of computing their energies and wave functions for any given pair of m and n. With this program, we perform CC(m)PT(n) and EOM-CC(m)PT(n) calculations of the ground-state energies and vertical excitation energies of selected small molecules for all possible values of m and 0⩽n⩽5. When the Hartree–Fock determinant is dominant in the FCI wave function, the second-order correction to CCSD [CC(2)PT(2)] reduces the differences in the ground-state energy between CCSD and FCI by more than a factor of 10, and thereby significantly outperforms CCSD(T) or even CCSDT. The third-order correction to CCSD [CC(2)PT(3)] further diminishes the energy difference between CC(2)PT(2) and FCI and its performance parallels that of some CCSD(TQ) models. CC(m)PT(n) for the ground state with some multideterminantal character and EOM-CC(m)PT(n) for the excitation energies, however, appear to be rather slowly convergent with respect to n.

Simultaneous optimization of exponents, centers of Gaussian-type basis functions, and geometry with full-configuration interaction wave function: Application to the ground and excited states of hydrogen molecule

We have extended the fully variational molecular orbital (FVMO) method to the full-configuration interaction (CI) wave function (full-CI FVMO). All variational parameters in the full-CI scheme, i.e., exponents and centers in Gaussian-type function (GTF) basis set, and nuclear positions, as well as the CI coefficients, are simultaneously optimized by using their analytical gradients. We have applied the full-CI FVMO method to the ground and electronic excited states of hydrogen molecule. In the ground state, the total energy (−1.174 015 hartree) and the internuclear distance (1.4016 bohr) obtained by the full-CI FVMO calculation with [8s4p2d] GTFs agree very well with the high-level calculation by the 249 term expansion in elliptic coordinates (−1.174 476 hartree and 1.4010 bohr, respectively). The excitation energies to the Σu+1, Πu1, Σg+3, and Πu3 Rydberg states calculated by the full-CI FVMO method with [8s4p2d] GTFs coincide with the experimental values within 52 cm−1. The present result can not be obtained with the conventional basis set approach because of the fact that our full-CI FVMO calculation gives an extremely accurate wave function with a relatively small number of basis functions owing to the extension of flexibility in the variational space.

Full configuration interaction and multiconfigurational spin density in boron and carbon atoms

The reliability of spin polarization method results for atomic spin densities, obtained with several widely used Gaussian basis sets, is examined by comparison with the results of full configuration interaction (FCI) calculations. The spin densities obtained with these basis sets using the spin polarization model and some other methods disagree with the FCI treatment. Since the FCI wave function is exact for a given basis, it is not clear that the spin polarization model will be generally reliable. A large active space multiconfigurational (CASSCF) calculation is shown to be inadequate as an alternative to FCI treatment. The importance of accounting at least to some extent for excitations to all orbitals in the complete space of basis functions is illustrated by very slow convergence of CASSCF results with increasing size of active space. The FCI results reported here can be used as benchmarks to test various approaches to spin density calculation.

Barrier for the H2CO→H2+CO reaction: A discrepancy between high-level electronic structure calculations and experiment

The unimolecular dissociation of formaldehyde to H2+CO was studied using extended basis set calculations and a variety of medium-to-high accuracy correlation recovery techniques. These included second and fourth order perturbation theory, multireference configuration interaction wave functions, coupled cluster theory with perturbative triples and full iterative triples, and estimated full configuration interaction wave functions. The intrinsic error of the electronic structure methods was assessed by extrapolating total energies to the complete basis set limit. Our best estimate of the barrier height, including zero point vibrational effects, is 81.9±0.3 kcal/mol, almost 3 kcal/mol larger than the experimental value of 79.2±0.8 kcal/mol. This estimate includes corrections for the effects of finite basis set truncation (which is negligible at the quintuple zeta level), higher order correlation recovery, core/valence correlation, and scalar relativistic effects. Using the same theoretical approach, we estimate the exothermicity of the dissociation reaction to be −1.6 kcal/mol, compared to experimental values in the −0.4 to −2.2 kcal/mol range. New calculations of the unimolecular dissociation rate constants using a variety of techniques failed to reconcile theory and experiment.

Ab initio/interpolated quantum dynamics on coupled electronic states with full configuration interaction wave functions

We present a new approach to first-principles molecular dynamics that combines a general and flexible interpolation method with ab initio evaluation of the potential energy surface. This hybrid approach extends significantly the domain of applicability of ab initio molecular dynamics. Use of interpolation significantly reduces the computational effort associated with the dynamics over most of the time scale of interest, while regions where potential energy surfaces are difficult to interpolate, for example near conical intersections, are treated by direct solution of the electronic Schrödinger equation during the dynamics. We demonstrate the concept through application to the nonadiabatic dynamics of collisional electronic quenching of Li(2p). Full configuration interaction is used to describe the wave functions of the ground and excited electronic states. The hybrid approach agrees well with full ab initio multiple spawning dynamics, while being more than an order of magnitude faster.

Correspondence between physical states and solutions to the coupled-cluster equations

To discuss the usefulness of the entries of the complete set of solutions to approximate coupled-cluster (CC) equations for the description of the states of many-electron systems, one has to be able to establish a correspondence between the states and solutions. The authors have explored four methods of finding links between the states (represented for the model by the full configuration interaction wave functions and energies) and solutions to the CC equations. Numerical studies for the P4 model, which belongs to the simplest realistic many-electron model systems, show that the states-solutions correspondence pattern strongly depends on the method employed. Only for a few states all methods considered have assigned the same solutions. Among these states are those that are energetically the closest to the reference determinants. For the remaining states the assignment is not unique and the accuracy of description depends on the physical features considered. The article is concluded with a comparison of the present results and results of the most recent studies on the structure of the complete sets of solutions to the CC equations with the findings and intuitions being the content of the work by Zivkovic and Monkhorst (J Math Phys 1978, 19, 1007).

A comparison between the CISD[TQ] wave function and other highly correlated methods: Molecular geometry and harmonic vibrational frequencies of MgH2

The linear MgH2 molecule was selected as a model system to compare the total electronic energy (ETOT), equilibrium bond length (re), and vibrational frequency (ωn) predictions from six different configuration interaction (CI) and coupled-cluster (CC) methods using three large correlation consistent basis sets. The three CI procedures employed incorporated all single and double substitutions (CISD) as well as all triple and quadruple substitutions (CISDTQ) or limited triple and quadruple substitutions (CISD[TQ]). The remaining three CC schemes included all single and double excitations (CCSD) as well as all triple excitations (CCSDT) or a perturbative approximation of the triple excitations [CCSD(T)]. Within the frozen core approximation employed in the study, the CISDTQ method constituted a full CI wave function. With the largest basis set this approach included 1.79 million configuration state functions and predicted re=1.711 Å, ω1=1602 cm−1, ω2=438 cm−1 and ω3=1628 cm−1. At the equilibrium geometry predicted by each method, agreement with the CISDTQ properties was observed to improve systematically in the following manner for all three basis sets: ETOT: CISD≪CISD[TQ]≈CCSD<CCSD(T)<CCSDT≈CISDTQ, re: CISD≪CCSD<CISD[TQ]<CCSD(T)<CCSDT≈CISDTQ, ωn: CISD≪CCSD<CISD[TQ] ≈CCSD(T)<CCSDT≈CISDTQ. With the largest basis set, ETOT was also computed after the Mg–H bond had been stretched to 3.0 Å. At this nuclear configuration the CISD[TQ] wave function outperformed the CC methods and recovered 99.8% full CI correlation energy while including over 100 times fewer configurations in the CI expansion. At this stretched geometry, agreement with the full CI correlation energy improved as follows: ETOT: CISD≪CCSD≪CCSD(T)≈CCSDT <CISD[TQ]≈CISDTQ.

Ab initio electronic structure of PtH+, PtH, Pt2, and Pt2H from a one-electron pseudopotential approach

A one-electron pseudopotential having an averaged d9 open shell in the core has been obtained for the ground state of Pt atom. With this pseudopotential we have studied the electronic structure of PtH+, PtH, Pt2, and Pt2H using large basis sets and an exact full configuration interaction wave function. In order to obtain reliable results, core–core repulsion, core–polarization and core–valence correlation effects have to be taken into account. These effects have been included through suitable core–core and core–polarization potentials. Comparison between existent and present calculations show that the one electron pseudopotential for Pt can be regarded as an average of the different real states arising from coupling of s valence shells with the open 5d9 levels of the d shell. Provided the extreme simplicity of the model, results obtained for small platinum containing systems are in fair agreement with previous highly sophisticated calculations and indicate that this one-electron pseudopotential for Pt can be effectively used to model surroundings of large surface cluster models or Pt complexes where d–d bonding is not dominant.

Fermi and Coulomb holes of molecules in excited states

AbstractCorrelation holes of electrons with the same (Fermi hole) and different (Coulomb hole) spins in the ground (X1Σ+), first (A1Σ+) and second (B1II) excited states of LiH were constructed from full configuration interaction (CI) wave functions. It was found that the shapes of both the Fermi and Coulomb holes in these states are dependent on the location of the reference electron. When the reference electron is chosen to be close to the Li nucleus, the Fermi correlation results in a large negative hole for all three states. However, the A1Σ+ excited state is further characterized by displaying a second hole around the H nucleus, and in the B1II state, the hole is elongated along the molecular axis. Coulomb correlation shows up strongly in the A1Σ+ state and, in addition, there is clearly correlation of electrons at the two nuclei. These features of the correlation holes were compared with those from a two‐Slater‐determinant model wave function. The Hartree, Fermi, and Coulomb screening potentials in these states were also studied in the light of possible modeling of the correlation functionals for the excited states. © 1995 John Wiley & Sons, Inc.

Generalized overlap amplitudes using the extended Koopmans’ theorem for Be

Approximate generalized overlap amplitudes (GOAs), also called Feynman–Dyson amplitudes, have been calculated from a full configuration interaction (CI) wave function for the ground state of beryllium using the extended Koopmans’ theorem (EKT). The GOAs were also calculated from the ground state CI wave function for Be and the ground state and excited state CI wave functions for Be+. The EKT GOAs are nearly identical to the corresponding CI GOAs for the lower 2S states of Be+ and for the Be+ 1s2s2 state which has a large GOA occupation number. There are many CI GOAs for which there is not a corresponding EKT GOA. This may be due in part to the limited size of the basis set and to the inability to include natural spin orbitals with small occupation numbers in the EKT calculations.