We perform a direct variational determination of the second-order (two-particle) density matrix corresponding to a many-electron system, under a restricted set of the two-index N-representability P-, Q-, and G-conditions. In addition, we impose a set of necessary constraints that the two-particle density matrix must be derivable from a doubly occupied many-electron wave function, i.e., a singlet wave function for which the Slater determinant decomposition only contains determinants in which spatial orbitals are doubly occupied. We rederive the two-index N-representability conditions first found by Weinhold and Wilson and apply them to various benchmark systems (linear hydrogen chains, He, N2, and CN(-)). This work is motivated by the fact that a doubly occupied many-electron wave function captures in many cases the bulk of the static correlation. Compared to the general case, the structure of doubly occupied two-particle density matrices causes the associate semidefinite program to have a very favorable scaling as L(3), where L is the number of spatial orbitals. Since the doubly occupied Hilbert space depends on the choice of the orbitals, variational calculation steps of the two-particle density matrix are interspersed with orbital-optimization steps (based on Jacobi rotations in the space of the spatial orbitals). We also point to the importance of symmetry breaking of the orbitals when performing calculations in a doubly occupied framework.We perform a direct variational determination of the second-order (two-particle) density matrix corresponding to a many-electron system, under a restricted set of the two-index N-representability P-, Q-, and G-conditions. In addition, we impose a set of necessary constraints that the two-particle density matrix must be derivable from a doubly occupied many-electron wave function, i.e., a singlet wave function for which the Slater determinant decomposition only contains determinants in which spatial orbitals are doubly occupied. We rederive the two-index N-representability conditions first found by Weinhold and Wilson and apply them to various benchmark systems (linear hydrogen chains, He, N2, and CN(-)). This work is motivated by the fact that a doubly occupied many-electron wave function captures in many cases the bulk of the static correlation. Compared to the general case, the structure of doubly occupied two-particle density matrices causes the associate semidefinite program to have a very favorable scaling as L(3), where L is the number of spatial orbitals. Since the doubly occupied Hilbert space depends on the choice of the orbitals, variational calculation steps of the two-particle density matrix are interspersed with orbital-optimization steps (based on Jacobi rotations in the space of the spatial orbitals). We also point to the importance of symmetry breaking of the orbitals when performing calculations in a doubly occupied framework.

Journal of Chemical Theory and ComputationJul 6, 201565 citations

Upper and Lower Bounds on Quantum-Mechanical Matrix Elements

Robert Blau, A. R. P. Rau, Larry Spruch

Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where $W$ is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the $n$ th and $m$ th states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for $W$ and only for real wave functions) or have been very conservative. We remove most of these restrictions ( $W$ need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on ${W}_{\mathrm{nm}}$ in terms of "simple" upper bounds (which may be poor) on ${{W}^{2}}_{\mathrm{nn}}$ . Here again, only for a very few particular choices of $W$ have such simple bounds on ${{W}^{2}}_{\mathrm{nn}}$ been given previously (for example, $n$ restricted to be the ground state, and $W$ the operator ${z}_{i}$ , the coordinate of the $i$ th electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators $W$ in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on ${{W}^{2}}_{\mathrm{nn}}$ do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of ${r}_{1}$ and ${r}_{1}^{2}$ ---we, therefore, require simple bounds on ${r}_{1}^{2}$ and ${r}_{1}^{4}$ ---for the states $1{s}^{2} ^{1}S$ and $1s2s ^{3}S$ of the helium atom, with rather good results.Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where $W$ is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the $n$th and $m$th states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for $W$ and only for real wave functions) or have been very conservative. We remove most of these restrictions ($W$ need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on ${W}_{\mathrm{nm}}$ in terms of "simple" upper bounds (which may be poor) on ${{W}^{2}}_{\mathrm{nn}}$. Here again, only for a very few particular choices of $W$ have such simple bounds on ${{W}^{2}}_{\mathrm{nn}}$ been given previously (for example, $n$ restricted to be the ground state, and $W$ the operator ${z}_{i}$, the coordinate of the $i$th electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators $W$ in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on ${{W}^{2}}_{\mathrm{nn}}$ do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of ${r}_{1}$ and ${r}_{1}^{2}$---we, therefore, require simple bounds on ${r}_{1}^{2}$ and ${r}_{1}^{4}$---for the states $1{s}^{2} ^{1}S$ and $1s2s ^{3}S$ of the helium atom, with rather good results.

Physical Review AJul 1, 197333 citations

Particle-number-conserving theory for nuclear pairing

Lu Jia

A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.

arXiv:1305.2697Physical Review COct 3, 201321 citations

Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where

W

is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the

n

th and

m

th states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for

W

and only for real wave functions) or have been very conservative. We remove most of these restrictions (

W

need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on

{W}_{\mathrm{nm}}

in terms of "simple" upper bounds (which may be poor) on

{{W}^{2}}_{\mathrm{nn}}

. Here again, only for a very few particular choices of

W

have such simple bounds on

{{W}^{2}}_{\mathrm{nn}}

been given previously (for example,

n

restricted to be the ground state, and

W

the operator

{z}_{i}

, the coordinate of the

i

th electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators

W

in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on

{{W}^{2}}_{\mathrm{nn}}

do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of

{r}_{1}

and

{r}_{1}^{2}

---we, therefore, require simple bounds on

{r}_{1}^{2}

and

{r}_{1}^{4}

---for the states $1{s}^{2} ^{1}S$ and

1s2s ^{3}S

of the helium atom, with rather good results.