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Physical Review

Antilinear Operators in Hartree-Bogolyubov Theory. I

Fedor Herbut, Milan Vujičić

Published Aug 20, 1968• 11 citations
Publisher

Abstract

Hartree-Bogolyubov (HB) theory is formulated in a basis-independent way, i.e., in terms of linear and antilinear operators acting in the one-particle space. For that purpose, some basic antilinear algebra is presented. The pairing tensor and the pairing potential are shown to represent two antilinear skew-Hermitian operators. The polar factorization of the first of them (the correlation operator ${\stackrel{^}{t}}_{a}$), i.e., ${\stackrel{^}{t}}_{a}={(\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}{\stackrel{^}{\ensuremath{\rho}}}^{2})}^{\frac{1}{2}}{\stackrel{^}{P}}_{a}$, shows that HB theory has only two variational (trial) operators: the density operator $\stackrel{^}{\ensuremath{\rho}}$ and the antilinear pairing operator ${\stackrel{^}{P}}_{a}$ which is defined by the properties ${{\stackrel{^}{P}}_{a}}^{+}={{\stackrel{^}{P}}_{a}}^{\ifmmode\dagger\else\textdagger\fi{}1}=\ensuremath{-}{\stackrel{^}{P}}_{a}$. These two operators commute. The former is the unique and very well-known variational operator of Hartree-Fock (HF) theory, and the latter represents a new variational freedom typical of HB theory. Most calculations, as for instance the Bardeen-Cooper-Schrieffer (BCS) approximation, restrict this freedom by choosing ${\stackrel{^}{P}}_{a}$ to be the time-reversal operator. The basic dynamical (Euler-Lagrange) equations of HB theory are obtained directly by varying linear and antilinear operators. They are expressed in a compact form, using only commutators and anticommutators of the kinematical and the dynamical operators: ${\stackrel{^}{A}}_{a}\ensuremath{\equiv}{[\stackrel{^}{h},{\stackrel{^}{t}}_{a}]}_{+}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}\frac{1}{2}]}_{+}=0, \stackrel{^}{B}\ensuremath{\equiv}{[\stackrel{^}{h},\stackrel{^}{\ensuremath{\rho}}]}_{\ensuremath{-}}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},{\stackrel{^}{t}}_{a}]}_{\ensuremath{-}}=0,$ where ${\ensuremath{\Delta}}_{a}$ is the pairing potential and $\stackrel{^}{h}$ is the one-particle Hamiltonian.Two identities are found between ${\stackrel{^}{A}}_{a}$ and $\stackrel{^}{B}$ which turn out to be very useful for obtaining solutions. Symmetries of the trial operators and of the solutions are discussed in great detail, special attention being paid to real HB solutions and their connection with some antiunitary symmetries. Several simple solutions are analyzed: (1) the case where $\stackrel{^}{\ensuremath{\rho}}$ is restricted to be a projector, i.e., ${\stackrel{^}{t}}_{a}=0$ (HF case); (2) the case where $\stackrel{^}{\ensuremath{\rho}}$ and ${\stackrel{^}{P}}_{a}$ are restricted to commute with a complete set of observables, which determines the eigensubspaces of $\stackrel{^}{\ensuremath{\rho}}$, and which, in particular cases of rotational and translational symmetries, fixes ${\stackrel{^}{P}}_{a}$ to be equal to the time-reversal operator (BCS case); and (3) the case where ${\stackrel{^}{P}}_{a}$ is any given symmetry operator of the Hamiltonian.

DOI
10.1103/physrev.172.1031

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Extended BCS Theories

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Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature $T$, the level density and the strength of the pairing force, are investigated analytically and numerically.

arXiv:nucl-th/9706041•Jun 17, 1997•0 citations

Zero-pairing and zero-temperature limits of finite-temperature Hartree-Fock-Bogoliubov theory

T. Duguet, Wouter Ryssens

Background: Recently, variational Hartree-Fock-Bogoliubov (HFB) mean-field equations were shown to possess a mathematically well-defined zero-pairing limit, independently of the closed- or open-shell character of the system under consideration. This limit is nontrivial for open-shell systems such that HFB theory does not reduce to the Hartree-Fock (HF) formalism in all cases.Purpose: The present work extends this analysis to finite-temperature HFB (FTHFB) theory by investigating the behavior of this more general formalism in the combined zero-temperature and zero-pairing limits.Methods: The zero-pairing and zero-temperature limits of the FTHFB statistical density operator constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction.Results: While the FTHFB density operator reduces to the projector corresponding to a pure HF Slater determinant for closed-shell nuclei, the FTHFB formalism does not reduce to the HF theory in all cases in the zero-temperature and zero-pairing limits, i.e., for open-shell nuclei. However, the fact that a nucleus can be of open-shell character in these joint limits is necessarily the result of some symmetry restrictions. Whenever it is the case, the nontrivial description obtained for open-shell systems is shown to depend on the order with which both limits are taken, i.e., the two limits do not commute for these systems. When the zero-temperature limit is performed first, the FTHFB density operator is demoted to a projector corresponding to a pure state made out of a linear combination of a finite number of Slater determinants with different (even) numbers of particles. When the zero-pairing limit is performed first, the FTHFB density operator remains a statistical mixture of a finite number of Slater determinants with both even and odd particle numbers. While the entropy (pairing density) is zero in the first (second) case, it does not vanish in the second (first) case in spite of the temperature (pairing) tending towards zero. The difference between both limits can have striking consequences for the (thermal) expectation values of observables. For instance, the particle-number variance does not vanish in either case and has limiting values that differ by a factor of two in both cases.Conclusions: While in the textbook situation associated with closed-shell nuclei Hartree-Fock-Bogoliubov (finite-temperature Hartree-Fock) theory reduces to Hartree-Fock theory in the zero-pairing (zero-temperature) limit, the present analysis demonstrates that a nontrivial and unexpected limit is obtained for this formalism in open-shell systems. This result sheds a new light on certain aspects of this otherwise very well-studied many-body formalism.

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Quantum Theory of Many-Particle Systems. II. Study of the Ordinary Hartree-Fock Approximation

Per‐Olov Löwdin

A system of $N$ antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.

Physical Review•Mar 15, 1955•467 citations

Zero-pairing limit of Hartree-Fock-Bogoliubov reference states

T. Duguet, B. Bally, A. Tichai

Background: The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. In this context, pairing correlations may be driven towards zero either on purpose via HFB calculations constrained on, e.g., the particle-number variance or simply because internucleon interactions cannot sustain pairing correlations in the first place in, e.g., closed-shell systems. While taking this limit constitutes a text-book problem when the system is of closed-shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system.Purpose: The present work demonstrates that the zero-pairing limit of an HFB state is well-defined independently of the average particle number $\mathrm{A}$ it is constrained to. Still, the nature of the limit state is shown to depend on the regime, i.e., on whether the nucleus characterizes as a closed-shell or an open-shell system when taking the limit. Finally, the consequences of the zero-pairing limit on Bogoliubov many-body perturbation theory (BMBPT) calculations performed on top of the HFB reference state are illustrated.Methods: The zero-pairing limit of a HFB state constrained to carry an arbitrary (integer) number of particles $\mathrm{A}$ on average is worked out analytically and realized numerically using a two-nucleon interaction derived within the frame of chiral effective field theory.Results: The zero-pairing limit of the HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number $\mathrm{A}$ of interest. First, the textbook situation is recovered for closed-shell systems, i.e., the limit state is reached for a finite value of the pairing strength (the well-known BardeenCooperSchrieffer (BCS) collapse) and takes the form of a single Slater determinant displaying (i) zero pairing energy, (ii) nondegenerate elementary excitations, and (iii) zero particle-number variance. Contrarily, a nonstandard situation is obtained for open-shell systems, i.e., the limit state is only reached for a zero value of the pairing interaction (no BCS collapse) and takes the form of a specific finite linear combination of Slater determinants displaying (a) a nonzero pairing energy, (b) degenerate elementary excitations, and (c) a nonzero particle-number variance for which an analytical formula is derived. This nonzero particle-number variance acts as a lower bound that depends in a specific way on the number of valence nucleons and on the degeneracy of the valence shell. All these findings are confirmed and illustrated numerically. Last but not least, BMBPT calculations of closed-shell (open-shell) nuclei are shown to be well-defined (ill-defined) in the zero-pairing limit.Conclusions: While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. In the present paper, the zero-pairing limit of a HFB state carrying an arbitrary number of particles has been worked out and shown to lead to drastic differences and consequences depending on the closed-shell or open-shell nature of the system in that limit. From a general perspective, the present analysis demonstrates that HFB theory does not reduce to HF theory when the pairing field is driven to zero in the HFB Hamiltonian matrix.

arXiv:2006.02871•Physical Review C•Nov 13, 2020•21 citations

Orbital Self-Consistent-Field Theory. I. General Theory for Multiconfigurational Wave Functions

William H. Adams

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Physical Review•Apr 5, 1967•44 citations

HARTREE--FOCK AND HARTREE--BOGOLYUBOV EQUATIONS WITH EXACT SYMMETRIES.

A. Friederich

•Jan 1, 1969•0 citations
Hartree-Bogolyubov (HB) theory is formulated in a basis-independent way, i.e., in terms of linear and antilinear operators acting in the one-particle space. For that purpose, some basic antilinear algebra is presented. The pairing tensor and the pairing potential are shown to represent two antilinear skew-Hermitian operators. The polar factorization of the first of them (the correlation operator {\stackrel{^}{t}}_{a}), i.e., {\stackrel{^}{t}}_{a}={(\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}{\stackrel{^}{\ensuremath{\rho}}}^{2})}^{\frac{1}{2}}{\stackrel{^}{P}}_{a}, shows that HB theory has only two variational (trial) operators: the density operator \stackrel{^}{\ensuremath{\rho}} and the antilinear pairing operator {\stackrel{^}{P}}_{a} which is defined by the properties {{\stackrel{^}{P}}_{a}}^{+}={{\stackrel{^}{P}}_{a}}^{\ifmmode\dagger\else\textdagger\fi{}1}=\ensuremath{-}{\stackrel{^}{P}}_{a}. These two operators commute. The former is the unique and very well-known variational operator of Hartree-Fock (HF) theory, and the latter represents a new variational freedom typical of HB theory. Most calculations, as for instance the Bardeen-Cooper-Schrieffer (BCS) approximation, restrict this freedom by choosing {\stackrel{^}{P}}_{a} to be the time-reversal operator. The basic dynamical (Euler-Lagrange) equations of HB theory are obtained directly by varying linear and antilinear operators. They are expressed in a compact form, using only commutators and anticommutators of the kinematical and the dynamical operators: {\stackrel{^}{A}}_{a}\ensuremath{\equiv}{[\stackrel{^}{h},{\stackrel{^}{t}}_{a}]}_{+}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}\frac{1}{2}]}_{+}=0, \stackrel{^}{B}\ensuremath{\equiv}{[\stackrel{^}{h},\stackrel{^}{\ensuremath{\rho}}]}_{\ensuremath{-}}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},{\stackrel{^}{t}}_{a}]}_{\ensuremath{-}}=0, where \ensuremathΔa{\ensuremath{\Delta}}_{a}\ensuremathΔa​ is the pairing potential and \stackrel{^}{h} is the one-particle Hamiltonian.Two identities are found between {\stackrel{^}{A}}_{a} and \stackrel{^}{B} which turn out to be very useful for obtaining solutions. Symmetries of the trial operators and of the solutions are discussed in great detail, special attention being paid to real HB solutions and their connection with some antiunitary symmetries. Several simple solutions are analyzed: (1) the case where \stackrel{^}{\ensuremath{\rho}} is restricted to be a projector, i.e., {\stackrel{^}{t}}_{a}=0 (HF case); (2) the case where \stackrel{^}{\ensuremath{\rho}} and {\stackrel{^}{P}}_{a} are restricted to commute with a complete set of observables, which determines the eigensubspaces of \stackrel{^}{\ensuremath{\rho}}, and which, in particular cases of rotational and translational symmetries, fixes {\stackrel{^}{P}}_{a} to be equal to the time-reversal operator (BCS case); and (3) the case where {\stackrel{^}{P}}_{a} is any given symmetry operator of the Hamiltonian.
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature TTT, the level density and the strength of the pairing force, are investigated analytically and numerically.
Background: Recently, variational Hartree-Fock-Bogoliubov (HFB) mean-field equations were shown to possess a mathematically well-defined zero-pairing limit, independently of the closed- or open-shell character of the system under consideration. This limit is nontrivial for open-shell systems such that HFB theory does not reduce to the Hartree-Fock (HF) formalism in all cases.Purpose: The present work extends this analysis to finite-temperature HFB (FTHFB) theory by investigating the behavior of this more general formalism in the combined zero-temperature and zero-pairing limits.Methods: The zero-pairing and zero-temperature limits of the FTHFB statistical density operator constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction.Results: While the FTHFB density operator reduces to the projector corresponding to a pure HF Slater determinant for closed-shell nuclei, the FTHFB formalism does not reduce to the HF theory in all cases in the zero-temperature and zero-pairing limits, i.e., for open-shell nuclei. However, the fact that a nucleus can be of open-shell character in these joint limits is necessarily the result of some symmetry restrictions. Whenever it is the case, the nontrivial description obtained for open-shell systems is shown to depend on the order with which both limits are taken, i.e., the two limits do not commute for these systems. When the zero-temperature limit is performed first, the FTHFB density operator is demoted to a projector corresponding to a pure state made out of a linear combination of a finite number of Slater determinants with different (even) numbers of particles. When the zero-pairing limit is performed first, the FTHFB density operator remains a statistical mixture of a finite number of Slater determinants with both even and odd particle numbers. While the entropy (pairing density) is zero in the first (second) case, it does not vanish in the second (first) case in spite of the temperature (pairing) tending towards zero. The difference between both limits can have striking consequences for the (thermal) expectation values of observables. For instance, the particle-number variance does not vanish in either case and has limiting values that differ by a factor of two in both cases.Conclusions: While in the textbook situation associated with closed-shell nuclei Hartree-Fock-Bogoliubov (finite-temperature Hartree-Fock) theory reduces to Hartree-Fock theory in the zero-pairing (zero-temperature) limit, the present analysis demonstrates that a nontrivial and unexpected limit is obtained for this formalism in open-shell systems. This result sheds a new light on certain aspects of this otherwise very well-studied many-body formalism.
A system of NNN antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
Background: The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. In this context, pairing correlations may be driven towards zero either on purpose via HFB calculations constrained on, e.g., the particle-number variance or simply because internucleon interactions cannot sustain pairing correlations in the first place in, e.g., closed-shell systems. While taking this limit constitutes a text-book problem when the system is of closed-shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system.Purpose: The present work demonstrates that the zero-pairing limit of an HFB state is well-defined independently of the average particle number A\mathrm{A}A it is constrained to. Still, the nature of the limit state is shown to depend on the regime, i.e., on whether the nucleus characterizes as a closed-shell or an open-shell system when taking the limit. Finally, the consequences of the zero-pairing limit on Bogoliubov many-body perturbation theory (BMBPT) calculations performed on top of the HFB reference state are illustrated.Methods: The zero-pairing limit of a HFB state constrained to carry an arbitrary (integer) number of particles A\mathrm{A}A on average is worked out analytically and realized numerically using a two-nucleon interaction derived within the frame of chiral effective field theory.Results: The zero-pairing limit of the HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number A\mathrm{A}A of interest. First, the textbook situation is recovered for closed-shell systems, i.e., the limit state is reached for a finite value of the pairing strength (the well-known BardeenCooperSchrieffer (BCS) collapse) and takes the form of a single Slater determinant displaying (i) zero pairing energy, (ii) nondegenerate elementary excitations, and (iii) zero particle-number variance. Contrarily, a nonstandard situation is obtained for open-shell systems, i.e., the limit state is only reached for a zero value of the pairing interaction (no BCS collapse) and takes the form of a specific finite linear combination of Slater determinants displaying (a) a nonzero pairing energy, (b) degenerate elementary excitations, and (c) a nonzero particle-number variance for which an analytical formula is derived. This nonzero particle-number variance acts as a lower bound that depends in a specific way on the number of valence nucleons and on the degeneracy of the valence shell. All these findings are confirmed and illustrated numerically. Last but not least, BMBPT calculations of closed-shell (open-shell) nuclei are shown to be well-defined (ill-defined) in the zero-pairing limit.Conclusions: While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. In the present paper, the zero-pairing limit of a HFB state carrying an arbitrary number of particles has been worked out and shown to lead to drastic differences and consequences depending on the closed-shell or open-shell nature of the system in that limit. From a general perspective, the present analysis demonstrates that HFB theory does not reduce to HF theory when the pairing field is driven to zero in the HFB Hamiltonian matrix.
A general formulation of multiconfigurational self-consistent-field theory is given. The fundamental condition to be satisfied by the MMM occupied orbitals ${\ensuremath{\psi}}_{\ensuremath{\mu}}$ of an $N(\ensuremath{\le}M)$-body system, in order that the total energy of one state be extremalized, is expressed in terms of the fundamental invariant $\mathbf{\ensuremath{\varrho}}=\ensuremath{\Sigma}{\ensuremath{\mu}=1}^{M}|{\ensuremath{\psi}}_{\ensuremath{\mu}}〉〈{\ensuremath{\psi}}_{\ensuremath{\mu}}|$. The condition placed on $\mathbf{\ensuremath{\varrho}}$ is of the same form as the condition on the fundamental invariant of the Hartree-Fock theory. Setting M=NM=NM=N, we derive the unrestricted Hartree-Fock equations from the condition on $\mathbf{\ensuremath{\varrho}}$. For M>NM>NM>N there are three freedoms one may take with the fundamental condition on $\mathbf{\ensuremath{\varrho}}$. Exploiting any or all of these freedoms yields alternative forms of the fundamental condition. This enables us to derive an effective one-body Hamiltonian which is the sum of a Hartree-type Hamiltonian and a correlation and exchange operator. For finite M>NM>NM>N, the one-body Hamiltonian contains a nonlocal exchange and correlation operator. This operator is defined in terms of the one- and two-body density matrices. The connection between the orbitals of this theory and those of the Hartree-Fock theory is explored. The theory as outlined here is applicable to any system of NNN identical particles, but our discussion is oriented towards electronic systems. The theory contains most self-consistent-field theories as special cases, and gives a basis for the self-consistent-field formulation of others.