Abstract
We consider a specific form of explicitly correlated Gaussians—with tensor pre-factors—which appear naturally when dealing with certain few-body systems in nuclear and particle physics. We derive analytic matrix elements with these Gaussians—overlap, kinetic energy, and Coulomb potential—to be used in variational calculations of those systems. We also perform a quick test of the derived formulae by applying them to p- and d-waves of the hydrogen atom.
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1 Introduction
Explicitly correlated Gaussians in various forms is a popular basis for variational calculations of quantum-mechanical few-body systems [1,2,3]. One important advantage of Gaussians is that their matrix elements are often analytic [2, 4,5,6,7] which greatly facilitates the calculations.
The most general Gaussians are shifted correlated Gaussians. However, they have a disadvantage: they lack a definite angular momentum. That might be problematic in applications where angular momentum is an important quantum number. In the literature one can find analytic formulae with Gaussians that acquire a definite angular momentum via pre-factors built of spherical harmonics [6]. However there exist applications in nuclear and particle physics, in particular with pions [8,9,10], where the variational basis-functions are multiplied by operators in the form of vectors and tensors.Footnote 1 These applications favour Gaussians with vector and tensor pre-factors rather than spherical harmonics. In this contribution we derive the analytic matrix elements with such Gaussians. We start with the matrix elements of the shifted correlated Gaussians, perform their Taylor expansion with respect to shift-vectors, and then collect the terms of specific orders, which ultimately gives the sought matrix elements.
Since some of the resulting formulae are rather long, particularly for the tensor pre-factor, we perform a quick test of the derived formulae by applying them to the lowest p- and d-wave states of the hydrogen atom.
2 Shifted Gaussians
Since we are going to use—as the starting point of our derivations—the analytic matrix elements with shifted correlated Gaussians we reproduce here the relevant analytic formulae from [7].
For an n-body system of particles the shifted correlated Gaussian with correlation matrix A and shift \(\textbf{a}\) is defined in coordinate space \(\textbf{r}\) as
where \({}^{\textsf{T}}\) denotes transposition, \(\textbf{r}\) is the column \(\{\vec {r}_1,\dots ,\vec {r}_n\}\) of the coordinates of the bodies, the shift \(\textbf{a}\) is a column of shift vectors \(\{\vec {a}_1,\dots ,\vec {a}_n\}\), and where
where the central dot denotes the scalar product of two vectors.
The overlap of two shifted Gaussians is given as
where \(R=(A+B)^{-1}\).
The matrix element of the kinetic energy operator is given as
where K is the (reduced) mass-matrix of the system of particles in the given set of coordinates.
The matrix element of the Coulomb potential is given as
where w is a given column of numbers \(\{w_1,\dots ,w_n\}\), and where \(\beta \doteq (w^{\textsf{T}}Rw)^{-1}\), \(\vec {q}\doteq \frac{1}{2} w^{\textsf{T}}R(\textbf{a}+\textbf{b})\).
3 Rank-0 Gaussians
The matrix elements with rank-0 (s-wave) Gaussians are well known. Notwithstanding, we reproduce here the relevant formulae for completeness. A rank-0 Gaussian is the zero-shift limit of a shifted Gaussian,
The overlap of two rank-0 Gaussians is the zero-shift limit of the shifted overlap (4),
Kinetic energy matrix element with rank-0 Gaussians is again given as the zero-shift limit of the corresponding shifted matrix element (5),
The Coulomb potential matrix element is, analogously, the zero-shift limit of the shifted matrix element (6),
4 Rank-1 Pre-factor Gaussians
The rank-1 pre-factor Gaussians are constructed by pre-factoring rank-0 Gaussians with the form \((\textbf{a}^{\textsf{T}}\textbf{r})\) (which represents a pure p-wave). We shall use the following notation,
where \(\textbf{a}\) is a column of polarization vectors \(\{\vec {a}_1,\dots ,\vec {a}_n\}\), and \(|(\textbf{a})A\rangle \) is our rank-1 Gaussian.
The rank-1 Gaussian can be obtained by collecting the linear term in the Taylor expansion of the shifted Gaussian in the shift-variable,
We shall denote this by the following notation,
The overlap of rank-1 Gaussians can be obtained by collecting the terms on the order \(O(\textbf{a}\textbf{b})\) from the Taylor expansions of the shifted overlap (and employing the symmetry of the matrix R),
The \(O(\textbf{a}\textbf{b})\) term of the expansion is given as
which leads to
Similarly, the kinetic energy matrix element with rank-1 Gaussians can be obtained by collecting the terms on the order \(O(\textbf{a}\textbf{b})\) from the Taylor expansion of the corresponding shifted Gaussian matrix element,
The Taylor expansion of the shifted kinetic energy matrix element (5) gives, term by term,
Summing up,
The Coulomb matrix element is again obtained by collecting the terms \(O(\textbf{a}\textbf{b})\) from the Taylor expansion of the shifted Coulomb matrix element (6),
We first expand the error-function,Footnote 2
where
Now, performing full expansion and collecting the terms \(O(\textbf{a}\textbf{b})\), gives
5 Rank-2 Pre-factor Gaussians
The rank-2 pre-factor Gaussians are constructed as
where \(\textbf{a}\) and \(\textbf{b}\) are the polarization vectors. The rank-2 pre-factor generally contains both s-waves and d-waves, however the condition \(\textbf{a}^{\textsf{T}}\textbf{b}=0\) eliminates the s-wave contribution.
The rank-2 Gaussian is the \(O(\textbf{a}\textbf{b})\) term in the Taylor expansion of the shifted Gaussian,
The overlap of rank-2 Gaussians is given by the terms \(O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})\) from the Taylor expansion of the shifted overlap,
Performing the Taylor expansion,
gives
5.1 Kinetic Energy
The rank-2 kinetic energy matrix element is given by the sum of the \(O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})\) terms in the Taylor expansion of the shifted Gaussian kinetic energy (5). Performing the Taylor expansion of (5) gives the following, term by term,
-
1.
the \(6~\textrm{Tr}(BKAR)M\) term,
$$\begin{aligned} 6~\textrm{Tr}(BKAR)M\underset{O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})}{\longrightarrow }6~\textrm{Tr}(BKAR)M_2 .\end{aligned}$$(34) -
2.
the \(\textbf{b}^{\textsf{T}}K\textbf{a}M\) term,
$$\begin{aligned}{} & {} (\textbf{c}+\textbf{d})^{\textsf{T}}K(\textbf{a}+\textbf{b}) e^{\frac{1}{4}(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})^{\textsf{T}}R(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})}M_0 \underset{O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})}{\longrightarrow }\nonumber \\{} & {} \quad \left[ (\textbf{a}^{\textsf{T}}K\textbf{c})(\textbf{b}^{\textsf{T}}R\textbf{d}) +(\textbf{a}^{\textsf{T}}K\textbf{d})(\textbf{b}^{\textsf{T}}R\textbf{c}) +(\textbf{b}^{\textsf{T}}K\textbf{c})(\textbf{a}^{\textsf{T}}R\textbf{d}) +(\textbf{b}^{\textsf{T}}K\textbf{d})(\textbf{a}^{\textsf{T}}R\textbf{c}) \right] \frac{1}{2} M_0 .\end{aligned}$$(35) -
3.
the \((\textbf{a}+\textbf{b})^{\textsf{T}}RBKAR (\textbf{a}+\textbf{b})M\) term,
$$\begin{aligned}{} & {} (\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})^{\textsf{T}}RBKAR(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d}) e^{\frac{1}{4}(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})^{\textsf{T}}R(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})}M_0 \underset{O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})}{\longrightarrow }\nonumber \\{} & {} \quad \left[ (\textbf{a}^{\textsf{T}}RBKAR \textbf{b})(\textbf{c} R\textbf{d}) +(\textbf{a}^{\textsf{T}}RBKAR \textbf{c})(\textbf{b} R\textbf{d}) +(\textbf{a}^{\textsf{T}}RBKAR \textbf{d})(\textbf{b} R\textbf{c}) \right] \frac{1}{2} M_0 \nonumber \\{} & {} \quad +\left[ (\textbf{b}^{\textsf{T}}RBKAR \textbf{a})(\textbf{c} R\textbf{d}) +(\textbf{b}^{\textsf{T}}RBKAR \textbf{c})(\textbf{a} R\textbf{d}) +(\textbf{b}^{\textsf{T}}RBKAR \textbf{d})(\textbf{a} R\textbf{c}) \right] \frac{1}{2} M_0 \nonumber \\{} & {} \quad +\left[ (\textbf{c}^{\textsf{T}}RBKAR \textbf{a})(\textbf{b} R\textbf{d}) +(\textbf{c}^{\textsf{T}}RBKAR \textbf{b})(\textbf{a} R\textbf{d}) +(\textbf{c}^{\textsf{T}}RBKAR \textbf{d})(\textbf{a} R\textbf{b}) \right] \frac{1}{2} M_0 \nonumber \\{} & {} \quad +\left[ (\textbf{d}^{\textsf{T}}RBKAR \textbf{a})(\textbf{b} R\textbf{c}) +(\textbf{d}^{\textsf{T}}RBKAR \textbf{b})(\textbf{a} R\textbf{c}) +(\textbf{d}^{\textsf{T}}RBKAR \textbf{c})(\textbf{a} R\textbf{b}) \right] \frac{1}{2} M_0 .\end{aligned}$$(36) -
4.
the \(-(\textbf{a}+\textbf{b})^{\textsf{T}}RBK \textbf{a}M\) term,
$$\begin{aligned}{} & {} -(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})^{\textsf{T}}RBK (\textbf{a}+\textbf{b}) e^{\frac{1}{4}(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})^{\textsf{T}}R(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})}M_0 \underset{O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})}{\longrightarrow }\nonumber \\{} & {} \quad -\left[ (\textbf{a}^{\textsf{T}}RBK \textbf{b})(\textbf{c}^{\textsf{T}}R\textbf{d}) +(\textbf{b}^{\textsf{T}}RBK \textbf{a})(\textbf{c}^{\textsf{T}}R\textbf{d}) \right] \frac{1}{2} M_0 \nonumber \\{} & {} \quad -\left[ (\textbf{c}^{\textsf{T}}RBK \textbf{a})(\textbf{b}^{\textsf{T}}R\textbf{d}) +(\textbf{c}^{\textsf{T}}RBK \textbf{b})(\textbf{a}^{\textsf{T}}R\textbf{d}) \right] \frac{1}{2} M_0 \nonumber \\{} & {} \quad -\left[ (\textbf{d}^{\textsf{T}}RBK \textbf{a})(\textbf{b}^{\textsf{T}}R\textbf{c}) +(\textbf{d}^{\textsf{T}}RBK \textbf{b})(\textbf{c}^{\textsf{T}}R\textbf{a}) \right] \frac{1}{2} M_0 .\end{aligned}$$(37) -
5.
the \(-\textbf{b}^{\textsf{T}}KAR (\textbf{a}+\textbf{b})M\) term,
$$\begin{aligned}{} & {} -(\textbf{c}+\textbf{d})^{\textsf{T}}KAR(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d}) e^{\frac{1}{4}(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})^{\textsf{T}}R(\textbf{a}+\textbf{b}+\textbf{c}+\textbf{d})}M_0 \underset{O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})}{\longrightarrow }\nonumber \\{} & {} \quad -\left[ (\textbf{c}^{\textsf{T}}KAR \textbf{a})(\textbf{b}^{\textsf{T}}R\textbf{d}) +(\textbf{c}^{\textsf{T}}KAR \textbf{b})(\textbf{a}^{\textsf{T}}R\textbf{d}) +(\textbf{c}^{\textsf{T}}KAR \textbf{d})(\textbf{a}^{\textsf{T}}R\textbf{b}) \right] \frac{1}{2} M_0 \nonumber \\{} & {} \quad -\left[ (\textbf{d}^{\textsf{T}}KAR \textbf{a})(\textbf{b}^{\textsf{T}}R\textbf{c}) +(\textbf{d}^{\textsf{T}}KAR \textbf{b})(\textbf{a}^{\textsf{T}}R\textbf{c}) +(\textbf{d}^{\textsf{T}}KAR \textbf{c})(\textbf{a}^{\textsf{T}}R\textbf{b}) \right] \frac{1}{2} M_0 .\end{aligned}$$(38)
5.2 Coulomb
The rank-2 Coulomb matrix element is given by the sum of the terms \(O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})\) in the Taylor expansion of the shifted Gaussian Coulomb matrix element (6),
where
Performing the expansion and collecting the \(O(\textbf{a}\textbf{b}\textbf{c}\textbf{d})\) terms gives
6 Symmetrization
The wave-function of a few-body system must be properly (anti)symmetrized under the exchange of the coordinates of identical particles. The total (anti)symmetrization can be done by a proper combination of the (elementary) permutation operators \({\hat{P}}\) that exchange the coordinates of the particles, \(\vec {r}_{i}\rightarrow \vec {r}_{i'}\), such that
where the matrix P has the matrix elements
Under this operation the terms \(\textbf{r}^{\textsf{T}}A\textbf{r}\) and \(\textbf{s}^{\textsf{T}}\textbf{r}\) transform as
Consequently the action of the permutation operator on the shifted/pre-factor Gaussians is given as
That is, under the permutation operation the Gaussians transform into the Gaussians of the same functional form—with the same analytic formulae for the matrix elements—only the correlation matrix and the shift/pre-factor vectors must be modified appropriately.
7 Quick Test
As a quick test of the derived formulae we calculate the lowest energies for s-, p-, and d-waves of the hydrogen atom by solving the corresponding Schrodinger equation,
where the Hamiltonian operator—kinetic energy plus Coulomb potential—is given (in Hartree units) as
The wave-function \(\Psi _l(\vec {r})\) with the given angular momentum l is represented as an expansion in terms of a size-n set of Gaussians with pre-factors corresponding to the given angular momentum,
where \(c_i^{(l)}\) are the expansion coefficients, and where \(|G^{(l)}\rangle \) is a Gaussian with a tensor pre-factor of rank-l.
The rank-0 (s-wave) Gaussians are given as
where \(\alpha _i\) is the variational parameter. The rank-1 (p-wave) Gaussians are taken as
where the polarization vector \(\vec {a}\) is chosen to be \(\vec {a}=\{0,0,1\}\) such that
The rank-2 Gaussians are taken in the form
where the polarization vectors are chosen as \(\vec {a}=\{1,0,0\}\), \(\vec {b}=\{0,1,0\}\) such that \(\vec {a}\cdot \vec {b}=0\) and the Gaussian represents a pure d-wave,
The Schrodinger equation in the space spanned by the given set of Gaussians is represented by the generalized matrix eigenvalue problem,
where \(N^{(l)}\) is the overlap matrix,
\(H^{(l)}\) is the Hamiltonian matrix,
and \(c^{(l)}=\{c_1^{(l)},\dots ,c_n^{(l)}\}\) is the column of the expansion coefficients. The matrix elements are calculated using the derived formulae.
The generalized matrix eigenvalue problem is solved using the standard method (via Cholesky decomposition of the overlap matrix). The parameters \(\alpha _i\) of the Gaussians are tuned to minimize the lowest eigenvalue by a gradient-descent method. The results of the calculations are shown on figure (7) where the lowest energies for s-, p-, and d-waves are shown as functions of the number of Guassians in the variational basis. The exact energiesFootnote 3 are reproduced within 4 decimal digits with the basis of 5 Gaussians.
8 Conclusion
We have derived analytic matrix elements—overlap, kinetic energy, and Coulomb potential—for correlated Gaussians with tensor pre-factors. Tensor pre-factor Gaussians might be useful in certain nuclear and particle physics applications, in particular when pions are included explicitly. We have done a quick test of the derived formulae by applying them to the p- and d-waves of the hydrogen atom.
Notes
For example, the operators \(\vec {\sigma }\vec {r}\) and \(\vec {\sigma }\vec {p}\), where \(\vec {\sigma }\) are Pauli matrices, \(\vec {r}\) and \(\vec {p}\) are the coordinate and momentum of the pion.
the exact energies are, in Hartee units, \(E_{l=0}=-\frac{1}{2}\), \(E_{l=1}=-\frac{1}{8}\), \(E_{l=2}=-\frac{1}{18}\).
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Fedorov, D.V., Teilmann, A.F., Østerlund, M.C. et al. Explicitly Correlated Gaussians with Tensor Pre-factors: Analytic Matrix Elements. Few-Body Syst 65, 75 (2024). https://doi.org/10.1007/s00601-024-01945-x
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DOI: https://doi.org/10.1007/s00601-024-01945-x