Off-diagonal Lagrange Multipliers Research Articles

On the orthogonality of states with approximate wavefunctions.

A variational solution to the eigenvalue problem for the Hamiltonian H, with orthogonality restrictions on eigenvectors of H to the vector H ∣ Φ0〉, where ∣Φ0〉 is an approximate ground-state vector, is proposed as a means to calculate excited states. The asymptotic projection (AP) method proposed previously is further developed and applied to solve this problem in a simple way. We demonstrate that the AP methodology does not require an evaluation of the matrix elements of operator H2, whereas conventional approaches-such as the elimination of off-diagonal Lagrange multipliers method, projection operator techniques, and other methods-do. It is shown, based on the results obtained for the single-electron molecular ions H2+, HeH2+, and H32+, that applying the new method to determine excited-state wavefunctions yields the upper bounds for excited-state energies. We demonstrate that regardless of whether the orthogonality constraint for states (〈Φ| Φ0〉 = 0) is applied, the zero-coupling constraint(〈Φ| H| Φ0〉 = 0) is imposed, or both of these restrictions are enforced simultaneously, practically the same excited-state energy is obtained if the basis set is almost complete. For the systems considered here, all schemes are capable of giving a sub-μhartree level of accuracy for the ground and excited states computed with different basis sets.

DBSR_HF: A B-spline Dirac–Hartree–Fock program

A B-spline version of a general Dirac–Hartree–Fock program is described. The usual differential equations are replaced by a set of generalized eigenvalue problems of the form (Ha−εaB)Pa=0, where Ha and B are the Hamiltonian and overlap matrices, respectively, and Pa is the two-component relativistic orbit in the B-spline basis. A default universal grid allows for flexible adjustment to different nuclear models. When two orthogonal orbitals are both varied, the energy must also be stationary with respect to orthonormal transformations. At such a stationary point the off-diagonal Lagrange multipliers may be eliminated through projection operators. The self-consistent field procedure exhibits excellent convergence. Several atomic states can be considered simultaneously, including some configuration-interaction calculations. The program provides several options for the treatment of Breit interaction and QED corrections. The information about atoms up to Z=104 is stored by the program. Along with a simple interface through command-line arguments, this information allows the user to run the program with minimal initial preparations. Program summaryProgram title: DBSR_HFCatalogue identifier: AEZK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 22643No. of bytes in distributed program, including test data, etc.: 354629Distribution format: tar.gzProgramming language: Fortran 95.Computer: No specific requirements to the computer.Operating system: Any system with a Fortran 95 compiler.Classification: 2.1.External routines: LAPACK (http://www.netlib.org/lapack/)Nature of problem:Relativistic Dirac–Hartree–Fock wavefunctions are determined for atoms in a bound state. These wavefunctions may be used to predict a variety of atomic properties.Solution method:The radial functions for large and small components of the one-electron spinor are expanded in B-spline bases. The variational principle applied to an energy functional that includes Lagrange multipliers for orthonormal constraints defines the Dirac–Hartree–Fock matrix for each orbital. Orthonormal transformations for a stationary solution were applied and Lagrange multipliers eliminated through projection operators.Restrictions:There is no restriction on calculations for the average or specific term energy of any atomic configuration with shells whose angular momenta are less than or equal to 9/2.Unusual features:The program allows the consideration of a few atomic states simultaneously. A simple interface through the command-line arguments allows the user to run the program with minimal initial preparations.Running time:From a few seconds to a few minutes depending on the atom under consideration.

A B-spline Hartree–Fock program

A B-spline version of a Hartree–Fock program is described. The usual differential equations are replaced by systems of non-linear equations and generalized eigenvalue problems of the form ( H a − ε a a B ) P a = 0 , where a designates the orbital. When orbital a is required to be orthogonal to a fixed orbital, this form assumes that a projection operator has been applied to eliminate the Lagrange multiplier. When two orthogonal orbitals are both varied, the energy must also be stationary with respect to orthogonal transformations. At such a stationary point, the matrix of Lagrange multipliers, ε a b = ( P b | H a | P a ) , is symmetric and the off-diagonal Lagrange multipliers may again be eliminated through projection operators. For multiply occupied shells, convergence problems are avoided by the use of a single-orbital Newton–Raphson method. A self-consistent field procedure based on these two possibilities exhibits excellent convergence. A Newton–Raphson method for updating all orbitals simultaneously has better numerical properties and a more rapid rate of convergence but requires more computer processing time. Both ground and excited states may be computed using a default universal grid. Output from a calculation for Al 3 s 2 3 p P 2 shows the improvement in accuracy that can be achieved by mapping results from low-order splines on a coarse grid to splines of higher order onto a refined grid. The program distribution contains output from additional test cases. Program summary Program title: SPHF version 1.00 Catalogue identifier: AEIJ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIJ_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 13 925 No. of bytes in distributed program, including test data, etc.: 714 254 Distribution format: tar.gz Programming language: Fortran 95 Computer: Any system with a Fortran 95 compiler. Tested on Intel Xeon CPU X5355, 2.66 GHz Operating system: Any system with a Fortran 95 compiler Classification: 2.1 External routines: LAPACK ( http://www.netlib.org/lapack/) Nature of problem: Non-relativistic Hartree–Fock wavefunctions are determined for atoms in a bound state that may be used to predict a variety atomic properties. Solution method: The radial functions are expanded in a B-spline basis [1]. The variational principle applied to an energy functional that includes Lagrange multipliers for orthonormal constraints defines the Hartree–Fock matrix for each orbital. Orthogonal transformations symmetrize the matrix of Lagrange multipliers and projection operators eliminate the off-diagonal Lagrange multipliers to yield a generalized eigenvalue problem. For multiply occupied shells, a single-orbital Newton–Raphson (NR) method is used to speed convergence with very little extra computation effort. In a final step, all orbitals are updated simultaneously by a Newton–Raphson method to improve numerical accuracy. Restrictions: There is no restriction on calculations for the average energy of a configuration. As in the earlier HF96 program [2], only one or two open shells are allowed when results are required for a specific LS coupling. These include: 1. ( nl ) N n ′ s , where l = 0 , 1 , 2 , 3 2. ( np ) N n ′ l , where l = 0 , 1 , 2 , 3 , … 3. ( nd ) ( n ′ f ) Unusual features: Unlike HF96, the present program is a Fortran 90/95 program without the use of COMMON. It is assumed that Lapack libraries are available. Running time: For Ac 7 s 2 7 p P 2 the execution time varied from 6.9 s to 9.1 s depending on the iteration method.

Ensemble-Hartree–Fock scheme for excited states. The optimized effective potential method

We review the ensemble-Hartree–Fock (eHF) scheme for excited states. The single-particle eHF equations contain different potentials for the various orbitals, leading to off-diagonal Lagrange multipliers that cannot be transformed away as in the ground state case. Using the optimized effective potential method we are able to construct a common local potential and the resulting theory is seen to describe accurately atomic excitation energies. A comparison of the eHF theory with the ensemble-Kohn–Sham (eKS) scheme suggests a correction for the ensemble exchange and correlation energy functional, that helps improve greatly the numerical results of the eKS scheme.

Basic theory of mass spectrometry

A B-spline version of a general Dirac–Hartree–Fock program is described. The usual differential equations are replaced by a set of generalized eigenvalue problems of the form (Ha−εaB)Pa=0, where Ha and B are the Hamiltonian and overlap matrices, respectively, and Pa is the two-component relativistic orbit in the B-spline basis. A default universal grid allows for flexible adjustment to different nuclear models. When two orthogonal orbitals are both varied, the energy must also be stationary with respect to orthonormal transformations. At such a stationary point the off-diagonal Lagrange multipliers may be eliminated through projection operators. The self-consistent field procedure exhibits excellent convergence. Several atomic states can be considered simultaneously, including some configuration-interaction calculations. The program provides several options for the treatment of Breit interaction and QED corrections. The information about atoms up to Z=104 is stored by the program. Along with a simple interface through command-line arguments, this information allows the user to run the program with minimal initial preparations.Program title: DBSR_HFCatalogue identifier: AEZK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 22643No. of bytes in distributed program, including test data, etc.: 354629Distribution format: tar.gzProgramming language: Fortran 95.Computer: No specific requirements to the computer.Operating system: Any system with a Fortran 95 compiler.Classification: 2.1.External routines: LAPACK (http://www.netlib.org/lapack/)Nature of problem:Relativistic Dirac–Hartree–Fock wavefunctions are determined for atoms in a bound state. These wavefunctions may be used to predict a variety of atomic properties.Solution method:The radial functions for large and small components of the one-electron spinor are expanded in B-spline bases. The variational principle applied to an energy functional that includes Lagrange multipliers for orthonormal constraints defines the Dirac–Hartree–Fock matrix for each orbital. Orthonormal transformations for a stationary solution were applied and Lagrange multipliers eliminated through projection operators.Restrictions:There is no restriction on calculations for the average or specific term energy of any atomic configuration with shells whose angular momenta are less than or equal to 9/2.Unusual features:The program allows the consideration of a few atomic states simultaneously. A simple interface through the command-line arguments allows the user to run the program with minimal initial preparations.Running time:From a few seconds to a few minutes depending on the atom under consideration.

The Hartree-Fock potential and its use in perturbation theory

Analytical expressions are given for the matrix elements of the UN-1 and UN-2 Hartree-Fock potentials used in perturbation theory to generate singly and doubly excited states. Formulae are derived for all off-diagonal Lagrange multipliers encountered. When correctly defined, UN-1 and UN-2 turn out to be so-called Silverstone-Huzinaga potentials, and therefore simultaneous orthogonality and a physically correct asymptotic behaviour of the virtual orbitals are ensured. The equations for the diagonal and off-diagonal matrix elements of the potential that underlie the theorems of Koopmans and Brillouin respectively, can directly be generalized from the UN-1 to the UN-2 case, although the modification of Koopmans' theorem is not trivial.

Localized and canonical atomic orbitals in self-interaction corrected local density functional approximation

The self-interaction corrected (SIC) version of the local spin density (LSD) approximation has been applied to the first two rows of the periodic table with particular emphasis on local orbital choice. These are the first SIC–LSD calculations for atomic systems that account for all nonspherical corrections and are based on a rigorous variational theory. The resulting total energies and orbital energies are improved in comparison to experiment and restore a desirable trend which is found in Hartree–Fock theory. We demonstrate that with a proper treatment of the SIC–LSD off-diagonal Lagrange multipliers, the viral theorem is satisfied at self-consistency.

Indirect exchange interaction between two doubly bridged paramagnetic atoms. a coupled-cluster model study

A model analysis is made of the influence of the ligands and the geometrical structure on the exchange-coupling constant, ecc, between two paramagnetic cations doubly bridged by two diamagnetic ligands. The model is based on a four-center, six-electron description of the superexchange unit, symmetry D 2h, with 1s slater type orbitals. A coupled-cluster method is employed. First, SCF orbitals are constructed for the unpaired electrons neglecting the interactions between the cations. Then, using these nonorthogonal orbitals, the direct exchange is evaluated. In the latter step, the influence of the ligands is gradually diminished and the results are compared with those from complete configuration interaction. It is found that the Coulomb and exchange ligand potentials are important in the evaluation of the ecc. Further, the cation-ligand-cation angle for which the ecc is equal to zero is not sensitive to variations of the ligand-admixture coefficient in the cation orbitals. The values of the ecc, however, do change essentially. Finally, the usual application of the Heitler-London treatment is revised. It is shown that the off-diagonal Lagrange multipliers in the SCF procedures introduce non-negligible contributions of the ligands to the ecc.

First order perturbation treatments for relativistic pseudopotentials and corrections to the Hartree-Fock method

The lowest order (1/c 2) perturbation theories recently developed [1, 2] to decribe the relativistic corrections to valence orbital energies in both Hartree-Fock and generalized Phillips-Kleinman pseudopotential theories are tested numerically for single valence electron atoms drawn for the entire periodic table. The corrections are analysed into contributions from the direct and indirect relativistic effects, the off-diagonal lagrange multipliers and the pseudopotential corrections. Theories in which the operators describing the indirect effect, which arises from relativistically induced modifications of the core charge density, are not exactly correct asymptotically and are shown to fail for atoms beyond the first row of the periodic table. Lowest order (1/c 2) perturbation theories in which the core potential is asymptotically exact are shown to give an excellent description of first and second row atoms, a useful description of the first transition series but to be only useful qualitatively for heavie...

Molecular MC–SCF calculations

AbstractA practical method for finding multi‐configurational SCF wave functions is proposed. The basic equation is equivalent to the Brillouin theorem; comparison with the usual SCF equations obtained through effective hamiltonians gives an interpretation of the offdiagonal Lagrange multipliers. Numerical applications to Formaldehyde in a minimum Slater‐type orbital basis with four different variational wave functions are reported. The molecular orbitals found in these calculations are localized on the chemical bonds. The largest contributions to the energy are obtained from π‐π and dispersion‐type σ‐π correlation.

The proper treatment of off-diagonal lagrange multipliers and coupling operators in self-consistent field equations

We show that past treatments of the off-diagonal Lagrange multipliers or coupling operators in the self-consistent field equations for open-shell systems and in the multi-configuration SCF equations are incomplete. In addition, we obtain the complete variational equations and show how these may be combined into one simple eigenvalue problem which is solved for all orbitals.

The orthogonality constrained basis set expansion method for treating off-diagonal lagrange multipliers in calculations of electronic wave functions

We propose an alternative to the coupling operator approach for the solution of the self-consistent field equations containing off-diagonal Lagrange multipliers. This method involves a transformation of the basis set after formation of the Hamiltonian matrices and leads to considerable simplification in the equations to be solved, reducing th ehomogeneous equations to pseudo-eigenvalue form. Typical results are reported for the lowest 2A 1 state of H 2O + and the lowest 3A 1 excited states of H 2O.

Configuration interaction in orbital theories

A systematic method is developed for estimating or calculating corrections for configuration interaction in atomic, molecular and nuclear wave-function calculations. Solutions of the Hartree-Fock equations for a single Slater determinant or approximate Hartree-Fock solutions obtained by Roothaan’s iterative procedure have special properties which are used to simplify the matrix of the many-particle Hamiltonian. A restricted self-consistent field method is proposed for treating states of low symmetry. This method avoids the off-diagonal Lagrange multipliers encountered in previous methods and is adapted to configuration interaction calculations.