Rational orbitals for two-electron S states

1981 ◽  
Vol 59 (10) ◽  
pp. 1552-1556
Author(s):  
F. W. Birss ◽  
W. den Hertog
Keyword(s):  
Wave Function ◽  
Configuration Interaction ◽  
Orthogonal Transformation ◽  
Open Shell ◽  
Wave Functions ◽  
Natural Orbitals ◽  
S States ◽  
Single Configuration

The concept of rational orbitals is introduced, based upon finding that pair of orbitals which yield the single configuration function which maximally overlaps with a configuration interaction wave function. They are simply obtained from the natural orbitals by an elementary orthogonal transformation and are more appropriate than natural orbitals to analysis of functions for open-shell states. The CI wave functions of a number of lS states of helium are analyzed and the nature of the rational orbitals investigated.

Extension of Frozen Natural Orbital Approximation to Open-Shell References: Theory, Implementation, and Application to Single-Molecule Magnets

2019 ◽  
Author(s):  
Pavel Pokhilko ◽  
Daniil Izmodenov ◽  
Anna I. Krylov
Keyword(s):  
Density Matrix ◽  
Single Molecule ◽  
Open Shell ◽  
State Density ◽  
Molecular Orbitals ◽  
The State ◽  
Wave Functions ◽  
Virtual Space ◽  
Single Molecule Magnets ◽  
Natural Orbitals

Natural orbitals are often used in quantum chemistry to achieve a more compact representation of correlated wave-functions. Using natural orbitals computed as eigenstates of the virtual-virtual block of the state density matrix instead of the canonical Hartree-Fock molecular orbitals results in smaller errors when the same fraction of virtual orbitals is frozen. This strategy, termed frozen natural orbitals (FNO) approach, has been successfully used to reduce the cost of state-specific coupled-cluster (CC) calculations, such as ground-state CC, as well as some multi-state methods, i.e., EOM-IP-CC (equation-of-motion CC method for ionization potentials). This contribution extends the FNO approach to the EOM-SF-CC ansatz (EOM-CC with spin-flip), which has been developed to describe certain multi-configurational wave-functions within the single-reference framework. In contrast to EOM-IP-CCSD, which describes open-shell target states by using a closed-shell reference, EOM-SF-CCSD relies on high-spin open-shell references (triplets, quartets, etc). Consequently, straightforward application of FNOs computed for an open-shell reference leads to an erratic behavior of the EOM-SF-CC energies and properties, which can be attributed to an inconsistent truncation of the α and β orbital spaces. A general solution to problems arising in the EOM-CC calculations utilizing open-shell references, termed OSFNO (open-shell FNO), is proposed. The OSFNO algo-rithm first identifies corresponding orbitals by means of singular value decomposition (SVD) of the overlap matrix of the α and β molecular orbitals and determines virtual orbitals corresponding to the singly occupied space. This is followed by SVD of the singlet part of the state density matrix in the remaining virtual orbital subspace. The so-computed FNOs preserve the spin purity of the open-shell orbital subspace to the extent allowed by the original reference thus facilitating a safe truncation of the virtual space. The performance of the OSFNO approximation in combination with different choices of reference orbitals is benchmarked for a set of diradicals and triradicals. For a set of di-copper single-molecule magnets, a conservative truncation criterion corresponding to a two-fold reduction of the virtual space in a triple-zeta basis leads to errors of 5–18 cm<sup>-1</sup> in the singlet–triplet gaps and errors of ∼2-3 cm<sup>-1</sup> in the spin–orbit coupling constants.

scholarly journals Stochastic Generalized Active Space Self-Consistent Field: Theory and Application

2021 ◽  
Author(s):  
Oskar Weser ◽  
Kai Guther ◽  
Khaldoon Ghanem ◽  
Giovanni Li Manni
Keyword(s):  
Wave Function ◽  
Configuration Interaction ◽  
Electron Correlation ◽  
Spin Exchange ◽  
Occupation Number ◽  
Heat Bath ◽  
Wave Functions ◽  
Density Matrices ◽  
Direct Evaluation ◽  
Active Space

An algorithm to perform stochastic generalized active space calculations, Stochastic-GAS, is presented, that uses the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver. Stochastic-GAS allows the construction and stochastic optimization of preselected truncated configuration interaction wave functions, either to reduce the computational costs of large active space wave function optimizations, or to probe the role of specific electron correlation pathways. As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations. Both local and cumulative minimum and maximum occupation number constraints are supported by Stochastic-GAS. The occupation number constraints are efficiently encoded in precomputed probability distributions, using the precomputed heat bath algorithm, which removes nearly all runtime overheads of GAS. This strategy effectively allows the FCIQMC dynamics to a priori exclude electronic configurations that are not allowed by GAS restrictions. Stochastic-GAS reduced density matrices are stochastically sampled, allowing orbital relaxations via Stochastic-GASSCF, and direct evaluation of properties that can be extracted from density matrices, such as the spin expectation value. Three test case applications have been chosen to demonstrate the flexibility of Stochastic-GAS: (a) the Stochastic-GASSCF optimization of a stack of five benzene molecules, that shows the applicability of Stochastic-GAS towards fragment-based chemical systems; (b) an uncontracted stochastic MRCISD calculation that correlates 96 electrons and 159 molecular orbitals, and uses a large (32, 34) active space reference wave function for an Fe(II)-porphyrin model system, showing how GAS can be applied to systematically recover dynamic electron correlation, and how in the specific case of the Fe(II)-porphyrin dynamic correlation further differentially stabilizes the triplet over the quintet spin state; (c) the study of an Fe4S4 cluster's spin-ladder energetics via highly truncated stochastic-GAS wave functions, where we show how GAS can be applied to understand the competing spin-exchange and charge-transfer correlating mechanisms in stabilizing different spin-states.

scholarly journals A study of some atomic properties for He-like selected ions

2007 ◽  
Vol 4 (2) ◽  
pp. 301-304
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Form Factor ◽  
Configuration Interaction ◽  
Diamagnetic Susceptibility ◽  
Wave Functions ◽  
Magnetic Shielding ◽  
Hartree Fock ◽  
Atomic Properties ◽  
Atomic Form Factor ◽  
He Atom

The atomic properties have been studied for He-like ions (He atom, Li+, Be2+ and B3+ions). These properties included, the atomic form factor f(S), electron density at the nucleus , nuclear magnetic shielding constant and diamagnetic susceptibility ,which are very important in the study of physical properties of the atoms and ions. For these purpose two types of the wave functions applied are used, the Hartree-Fock (HF) waves function (uncorrelated) and the Configuration interaction (CI) wave function (correlated). All the results and the behaviors obtained in this work have been discussed, interpreted and compared with those previously obtained.

The effect of configuration interaction on the low terms of the spectra of oxygen

1937 ◽  
Vol 33 (2) ◽  
pp. 240-249 ◽  
Author(s):  
D. R. Hartree ◽  
Bertha Swirles
Keyword(s):  
Wave Function ◽  
General Theory ◽  
Configuration Interaction ◽  
Matrix Elements ◽  
Atomic Spectra ◽  
Single Configuration

The ratios of the inter-multiplet separations for the lowest states of O, O+and O++obtained by Slater's method depart considerably from the observed values. In this method it is assumed that matrix elements of the Hamiltonian involving two different configurations are negligible, so that each state can be described by a single configuration, whereas these matrix elements are probably appreciable, and a better approximation is obtained by use of a wave function corresponding to a superposition of more than one configuration. The effect of this superposition of configurations has been called “configuration interaction”, and the general theory of it is discussed in Condon and Shortley'sTheory of atomic spectra, Chap. xv. It is shown that it occurs only between terms of the sameLandS, and of the same parity (∑leven or odd). Few quantitative applications, however, have yet been made. A calculation by Bacher for Mg shows that the effect can be considerable although the states are quite widely separated.

scholarly journals Evaluation of the one electron expeetation values for different wave functions

2004 ◽  
Vol 1 (2) ◽  
pp. 336-339
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Slater Determinant ◽  
Open Shell ◽  
Wave Functions ◽  
Expectation Values ◽  
Electronic Density ◽  
Shell System ◽  
Positioning Technique ◽  
The One ◽  
Open Shell System

The aim of this work is to evaluate the onc-electron expectation values < r > from the radial electronic density funetion D(r) for different wave ?'unctions for the 2s state of Li atom. The wave functions used were published in 1963,174? and 1993 , respectavily. Using " " ' wave function as a Slater determinant has used the positioning technique for the analysis open shell system of Li (Is2 2s) State.

Extension of Frozen Natural Orbital Approximation to Open-Shell References: Theory, Implementation, and Application to Single-Molecule Magnets

2019 ◽  
Author(s):  
Pavel Pokhilko ◽  
Daniil Izmodenov ◽  
Anna I. Krylov
Keyword(s):  
Density Matrix ◽  
Single Molecule ◽  
Open Shell ◽  
State Density ◽  
Molecular Orbitals ◽  
The State ◽  
Wave Functions ◽  
Virtual Space ◽  
Single Molecule Magnets ◽  
Natural Orbitals

Natural orbitals are often used in quantum chemistry to achieve a more compact representation of correlated wave-functions. Using natural orbitals computed as eigenstates of the virtual-virtual block of the state density matrix instead of the canonical Hartree-Fock molecular orbitals results in smaller errors when the same fraction of virtual orbitals is frozen. This strategy, termed frozen natural orbitals (FNO) approach, has been successfully used to reduce the cost of state-specific coupled-cluster (CC) calculations, such as ground-state CC, as well as some multi-state methods, i.e., EOM-IP-CC (equation-of-motion CC method for ionization potentials). This contribution extends the FNO approach to the EOM-SF-CC ansatz (EOM-CC with spin-flip), which has been developed to describe certain multi-configurational wave-functions within the single-reference framework. In contrast to EOM-IP-CCSD, which describes open-shell target states by using a closed-shell reference, EOM-SF-CCSD relies on high-spin open-shell references (triplets, quartets, etc). Consequently, straightforward application of FNOs computed for an open-shell reference leads to an erratic behavior of the EOM-SF-CC energies and properties, which can be attributed to an inconsistent truncation of the α and β orbital spaces. A general solution to problems arising in the EOM-CC calculations utilizing open-shell references, termed OSFNO (open-shell FNO), is proposed. The OSFNO algo-rithm first identifies corresponding orbitals by means of singular value decomposition (SVD) of the overlap matrix of the α and β molecular orbitals and determines virtual orbitals corresponding to the singly occupied space. This is followed by SVD of the singlet part of the state density matrix in the remaining virtual orbital subspace. The so-computed FNOs preserve the spin purity of the open-shell orbital subspace to the extent allowed by the original reference thus facilitating a safe truncation of the virtual space. The performance of the OSFNO approximation in combination with different choices of reference orbitals is benchmarked for a set of diradicals and triradicals. For a set of di-copper single-molecule magnets, a conservative truncation criterion corresponding to a two-fold reduction of the virtual space in a triple-zeta basis leads to errors of 5–18 cm<sup>-1</sup> in the singlet–triplet gaps and errors of ∼2-3 cm<sup>-1</sup> in the spin–orbit coupling constants.

scholarly journals Stochastic Generalized Active Space Self-Consistent Field: Theory and Application

2021 ◽  
Author(s):  
Oskar Weser ◽  
Kai Guther ◽  
Khaldoon Ghanem ◽  
Giovanni Li Manni
Keyword(s):  
Wave Function ◽  
Configuration Interaction ◽  
Electron Correlation ◽  
Spin Exchange ◽  
Occupation Number ◽  
Heat Bath ◽  
Wave Functions ◽  
Density Matrices ◽  
Direct Evaluation ◽  
Active Space

An algorithm to perform stochastic generalized active space calculations, Stochastic-GAS, is presented, that uses the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver. Stochastic-GAS allows the construction and stochastic optimization of preselected truncated configuration interaction wave functions, either to reduce the computational costs of large active space wave function optimizations, or to probe the role of specific electron correlation pathways. As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations. Both local and cumulative minimum and maximum occupation number constraints are supported by Stochastic-GAS. The occupation number constraints are efficiently encoded in precomputed probability distributions, using the precomputed heat bath algorithm, which removes nearly all runtime overheads of GAS. This strategy effectively allows the FCIQMC dynamics to a priori exclude electronic configurations that are not allowed by GAS restrictions. Stochastic-GAS reduced density matrices are stochastically sampled, allowing orbital relaxations via Stochastic-GASSCF, and direct evaluation of properties that can be extracted from density matrices, such as the spin expectation value. Three test case applications have been chosen to demonstrate the flexibility of Stochastic-GAS: (a) the Stochastic-GASSCF optimization of a stack of five benzene molecules, that shows the applicability of Stochastic-GAS towards fragment-based chemical systems; (b) an uncontracted stochastic MRCISD calculation that correlates 96 electrons and 159 molecular orbitals, and uses a large (32, 34) active space reference wave function for an Fe(II)-porphyrin model system, showing how GAS can be applied to systematically recover dynamic electron correlation, and how in the specific case of the Fe(II)-porphyrin dynamic correlation further differentially stabilizes the triplet over the quintet spin state; (c) the study of an Fe4S4 cluster's spin-ladder energetics via highly truncated stochastic-GAS wave functions, where we show how GAS can be applied to understand the competing spin-exchange and charge-transfer correlating mechanisms in stabilizing different spin-states.

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