Approximate Natural Orbitals Research Articles

Approximate natural orbitals via the biorthogonal valence bond method

A method is introduced for determining an initial approximation to the first-order density matrix and natural orbitals of a molecular system within the valence bond formalism. The nonorthogonality problem usually associated with these calculations is avoided by using a dual biorthogonal basis set. Results of test calculations are presented and the feasibility of using this method in large-scale natural orbital CI calculations is discussed.

Iterative natural orbitals for configuration interaction using perturbation theory

Approximate natural orbitals are determined iteratively from CI expansions constructed using first-order perturbation theory in order to investigate the possibility of eliminating the complete transformation of MO integrals on each iteration. Results on LiH and H2O are compared with fully variationally determined NO's to assess questions of convergence.

Many-body calculations of the hyperfine interaction of some excited states of alkali atoms, using approximate Brueckner or natural orbitals

The theory of maximum-overlap or Brueckner orbitals and of natural Orbitals is reviewed for closed-shell systems. A technique is described to calculate approximate Brueckner or natural orbitals for systems with a single valence electron, and this is applied together with the linked-diagram expansion of effective operators to evaluate the hyperfine interaction of some excited states of alkali atoms. It is found that it is in this way possible to reproduce reasonably well also the highly perturbed interactions of the excitedd states, which is not possible in a low-order expansion based on HF orbitals. Numerical results are given for the 5d state in K and the 4d state in Rb, where good experimental information is available.

Theoretical study of the vertical electronic spectrum of O 2: Mixing of valence and Rydberg states

All-valence-electron CM calculations are reported for a large number of electronic states of O 2 at the ground state equilibrium bond length. The configuration subspaces considered include all single and double excitations with respect to a series of the most important terms in the expansion of each state. The importance of the choice of such reference configurations as well as of the use of approximate natural orbitals in these calculations is discussed Mixing at Rydberg and valence states is observed in numerous cases and the significance of this phenomenon in the interpretation of the electrons spectrum of this system is considered.

Individualized configuration selection in CI calculations with subsequent energy extrapolation

A configuration selection method for CI calculations is discussed and applied in which the energy lowering produced in a secular equation by the addition of a given test species to a series of dominant configurations is used as an ordering parameter. Configurations with energy lowerings below a given energy cut-off value are not included in the final secular equations but instead a method of estimating the combined effect of the neglected species on the corresponding non-selected CI results is developed. The influence of the choice of main configurations used in the selection process is given close examination as well as the importance of the MO basis employed in the treatment as a whole; in the latter case a non-iterative procedure for obtaining approximate natural orbitals for such calculations is suggested. The resulting configuration selection procedure is equally applicable to all types of electronic states in any nuclear geometry and the results of the associated CI calculations are seen to be essentially equivalent to a complete treatment in which all single- and double-excitation species with respect to aseries of dominant configurations in a given state are included.

MC–SCF and CI calculations for the ammonia molecule

Multiconfiguration self-consistent field calculations are reported for the ammonia molecule, both with and without symmetry restrictions. Correlation energies of 0.039 and 0.080 a.u. have been obtained, respectively, for the core and the valence shell (without the symmetry restriction). It is found that the MC–SCF method is not stable with respect to the symmetry constraint, as the result of a better description of the left-right correlation. A comparison is made of configuration interaction calculations based either on canonical SCF or MC–SCF or approximate natural orbitals. Comparable values of the correlation energy are obtained from CI calculations which are based on the same number of MC–SCF orbitals or approximate natural orbitals. Analysis in terms of pair correlation energies shows that the approximate natural orbitals achieve less of the intrapair and more of the interpair correlation energy than the MC–SCF orbitals. Strongly occupied natural orbitals are close to the corresponding SCF orbitals, both being more delocalized than the MC–SCF orbitals. The weakly occupied NO and MC–SCF orbitals are comparatively much closer.

Genäherte natürliche Orbitale für die Grundzustände von Li und Li-; Elektronenaffinität von Li- / Approximate natural orbitals for the groundstates of li and li-; electron-affinity of li-

Abstract Starting from a set of Slater-type atomic orbitals with seven s-, four p-, one d- and one f-type functions the total wavefunction is constructed by a C I-calculation. The highest degree of the secular equation is 84 for Li (2S) and 142 for Li- (1S). By diagonalizing the first-order density matrices approximate natural orbitals are obtained. These are analysed and used again in a further C I-calculation. The best values for the electron-affinity are 0.572 eV and 0.614 eV. For the latter value only the correlaton energy of the L-shell of Li- is considered

Natural orbital iterations for the ground state of nitric oxide

AbstractA method for the use of natural orbital iterations in limited CI calculations is discussed. This method is then applied to the ground X2II state of the nitric oxide molecule at its experimental equilibrium internuclear separation to yield the total energy, dipole moment, spin densities at each nucleus, and approximate natural spin orbitals for this molecule.

Studies in Configuration Interaction: The First-Row Diatomic Hydrides

Molecular properties of the first-row diatomic hydrides, calculated with accurate configuration-interaction wave functions, are presented. Ground-state functions, constructed from approximate natural orbitals, were determined at the equilibrium internuclear separation for each molecule. The basis sets used, were capable of reproducing recently published self-consistent-field energies to within 0.003 hartrees. A minimum of 70% of the correlation energy and an error in the dipole moment of less than 3% were obtained in a systematic study of each molecule. To attain this accuracy, large numbers of singly and doubly excited configurations were used in the wave functions.

Direct Calculation of Approximate Natural Orbitals and Natural Expansion Coefficients of Atomic and Molecular Electronic Wavefunctions. II. Decoupling of the Pair Equations and Calculation of the Pair Correlation Energies for the Be and LiH Ground States

Starting from the integro-differential equations satisfied by the natural orbitals (NO's) in the framework of the antisymmetrized-product-of-strongly-orthogonal-geminals (APSG) approximation, a set of decoupled equations for the different electron pairs is derived, which corresponds to the physical model of independent electron pairs in the Hartree-Fock field of the other electrons. These simplified equations lead to an easily practicable computational scheme which furnishes directly the correlation energy of the different pairs and a wavefunction, which is an antisymmetrized product of geminals which are, however, no longer strongly orthogonal in the rigorous sense. The total intrapair correlation energy, which is defined as the energy expectation value (with respect to this wavefunction) minus the Hartree-Fock energy, differs from the sum of the pair correlation energies by a correction term, the magnitude of which depends on the highest occupation number of a weakly occupied NO. In the Be ground state, where this highest occupation number is about 0.1, the correction term amounts to ∼2% of the correlation energy; in the LiH ground state, none of these occupation numbers is bigger than 0.01 and the correction term is much less than 1% of the correlation energy. A detailed discussion of these correction terms, which is of vital importance for all theories of decoupled electron pairs, is given in the Appendix. An alternative, simpler way to get a rigorous bound for the correlation energy, based on a configuration interaction calculated only with doubly substituted NO configurations, is also indicated. The method is applied numerically to the Be and LiH ground states, and it shows quite promising results. In Be, the dependence of the intrapair correlation energy on the “localization” of the strongly occupied orbitals is examined in detail; so is the influence of occupied outer-shell electrons on the correlation energy of the inner one. Different previous calculations on the Be ground state are compared with ours, and the formal relation between these schemes is pointed out. In the LiH calculation, it is shown that the weakly occupied approximate natural orbitals, as calculated by our scheme, are localized in the same region of space as the corresponding strongly occupied ones. The rather poor result for the correlation energy of the inner pair (0.0302 a.u. rather than 0.0414 a.u. in the Li atom) is due to the inadequacy of the basis of pure Gaussians used; the basis is much better for the outer pair; 78% of the exact overall correlation energy is accounted for. A potential curve, including correlation, is calculated for the LiH molecule. The change in correlation energy is essentially that of the valence-shell correlation energy. For this, the σ-correlation energy increases with distance, and the π-correlation energy decreases, so that near the minimum, the total correlation energy remains almost constant. Twenty minutes of IBM 7094 Model 1 were necessary for one point of the potential curve.

The hartree-fock and the correlation energies of the H +3 ion and their dependence on the nuclear configuration

The method of solving the two-electron Schrödinger equation numerically by ??? calculation of approximate natural orbitals is applied to the H + 3 ion for different nuclear configurations, using a basis of Gaussian lobe functions. For the equilateral triangular equilibrium configuration and the lowest-energy linear configuration total energies of −1.3359 and −1.2729 a.u. respectively are obtained. the accurate eigenvalues can be estimated to be −1.341 ± 0.001 and −1.278 ± 0.001 a.u. Most striking is the change in the correlation energy, which has been calculated to be −0.039 a.u. in the triangular and −0.047 a.u. in the linear configuration.

Approximate natural orbitals for carbon1 S

Approximate natural orbitals for carbon1 S

Different Orbitals for Different Spins. Singlet S Ground State of Helium

The ``different-orbitals-for-different-spins'' (DODS) wavefunction for the ground state of He is expressed in terms of Shull and Löwdin basis orbitals. The energy and the occupation amplitudes of the two approximate natural orbitals have been obtained by solving two coupled pseudoeigenvalue matrix equations. The DODS wavefunction accounts for 94% of the radial correlation energy.

On the solution of the quantum mechanical two-electron problem by direct calculation of the natural orbitals

The wave function of the ground state of the H2 molecule is calculated directly in its natural expansion form, the approximate natural orbitals (NO) being expressed as linear combinations of Slater type functions centered at the midpoint of the molecule. One obtains as total energy (in the equilibrium distance) − 1.168 a.u. (exact − 1.174 a.u.) which seems to be the best one-center result for H2 known so far; 96% of the binding energy is accounted for. The accuracy of this approach is limited due to the rather slow convergency of the onecenter expansion of the orbitals. The Hartree-Fock energy calculated with the same basis (−1.129 a.u.) is about as much in error with respect to the exact HF energy (− 1.136 a.u.) as the energy of the NO expansion with respect to the experimental one.

Approximate Natural Orbitals for Four-Electron Systems

Approximate natural orbitals for four-electron wavefunctions for Be as calculated by Boys, Matsen, Watson, and Weiss and LiH as calculated by Ebbing are obtained by diagonalization of the appropriate first-order reduced density matrices. The formalism for carrying out the process is extended to the case of a nonorthogonal basis. Occupation numbers are presented and compared for the functions reported. In addition, truncated natural expansions reproducing the original functions are tabulated. Natural-orbital functions are tabulated. Natural-orbital occupation numbers, the orbitals themselves, and the natural expansions are shown to have considerable value in comparing these functions. The nature of the approximations used in the individual functions is clarified by the analysis, and considerable insight is afforded to the nature of the physical system.

Approximate natural spin orbitals for theBstate of the hydrogen molecule

Approximate natural spin orbitals for the<i>B</i>state of the hydrogen molecule

Approximate Natural Spin Orbitals for the Hydrogen Molecule

Approximate natural spin orbitals and the corresponding occupation numbers are obtained over a wide range of internuclear separations for the Σg+1 ground state and the Σu+3 excited state of the hydrogen molecule, using the Hirschfelder-Linnett wave functions. The resulting natural expansions and their variation with internuclear separation are then discussed in terms of electronic configurations.