Abstract
The nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) approach is extended and applied to the positronic systems PsH, LiPs, and e^{+}LiH. In this implementation, all electrons and positrons are treated quantum mechanically, and all nuclei are treated classically. This approach utilizes molecular orbital techniques with Gaussian basis sets for the electrons and positrons and includes electron-positroncorrelation with explicitly correlated Gaussian-type geminal functions. An efficient strategy is developed to reduce the number of variational parameters in the NEO-XCHF calculations. The annihilation rates, electron and positron densities, and electron-positron contact densities are compared to available results from higher-level calculations. Our analysis illustrates that the NEO-XCHF method produces qualitative to semi-quantitative results for these properties at a relatively low computational cost by treating only the essential electron-positroncorrelation explicitly. The NEO-HF method, which does not include explicit correlation and therefore is extremely efficient, is found to provide qualitatively accurate electron-positron contact densities for the e^{+}LiH system but not for the LiPs system. Thus, the utility of the NEO-HF method for determining where annihilation occurs is system dependent and not generally reliable. The NEO-XCHF method, however, provides a computationally practical and reliable approach for determining where annihilation will occur in positronic systems.
We thank Andrew Sirjoosingh for helpful discussions. We gratefully acknowledge funding from (U.S.) Air Force Office of Scientific Research (AFOSR) Grant No. FA9550-10-1-0081.
I. INTRODUCTION
II. THEORY AND METHODS
III. RESULTS AND DISCUSSION
A. Positronium hydride
B. Lithium positride
C. e^{+}LiH
IV. CONCLUSIONS
Key Topics
- Positrons
- 80.0
- Electron correlation calculations
- 34.0
- Electron positron interactions
- 33.0
- Wave functions
- 19.0
- Basis sets
- 11.0
Figures
Dependence of the PsH NEO-XCHF radial (a) electron density, (b) positron density, and (c) electron-positron contact density on the number of terms in the GTG expansion for N _{gem} = 0−8. The dashed lines denote NEO-HF results, and the solid lines denote NEO-XCHF results for varying values of N _{gem}. For electron density, the solid lines for different values of N _{gem} are indistinguishable. For the positron and contact densities, increasing the value of N _{gem} increases the magnitude of the maximum until convergence at N _{gem} = 2 for the positron density and N _{gem} ≈ 7 for the contact density.
Dependence of the PsH NEO-XCHF radial (a) electron density, (b) positron density, and (c) electron-positron contact density on the number of terms in the GTG expansion for N _{gem} = 0−8. The dashed lines denote NEO-HF results, and the solid lines denote NEO-XCHF results for varying values of N _{gem}. For electron density, the solid lines for different values of N _{gem} are indistinguishable. For the positron and contact densities, increasing the value of N _{gem} increases the magnitude of the maximum until convergence at N _{gem} = 2 for the positron density and N _{gem} ≈ 7 for the contact density.
Dependence of the PsH NEO-XCHF radial electron-positron intracule density on the number of terms in the GTG expansion for N _{gem} = 0−8. (a) I _{ ep } without integration over angular coordinates and (b) I _{ ep } with integration over angular coordinates (i.e., multiplication of part (a) by 4πr ^{2} for this spherically symmetric system). The dashed lines denote NEO-HF results, and the solid lines denote NEO-XCHF results for varying values of N _{gem}, where increasing the value of N _{gem} increases the magnitude of the maximum until convergence.
Dependence of the PsH NEO-XCHF radial electron-positron intracule density on the number of terms in the GTG expansion for N _{gem} = 0−8. (a) I _{ ep } without integration over angular coordinates and (b) I _{ ep } with integration over angular coordinates (i.e., multiplication of part (a) by 4πr ^{2} for this spherically symmetric system). The dashed lines denote NEO-HF results, and the solid lines denote NEO-XCHF results for varying values of N _{gem}, where increasing the value of N _{gem} increases the magnitude of the maximum until convergence.
Comparison of the radial PsH electron density (blue), positron density (red), and electron-positron contact density (black) calculated with the NEO-XCHF (solid lines) and ECG^{44} (dashed lines) methods. Contact densities have been scaled by a factor of 10.
Comparison of the radial PsH electron density (blue), positron density (red), and electron-positron contact density (black) calculated with the NEO-XCHF (solid lines) and ECG^{44} (dashed lines) methods. Contact densities have been scaled by a factor of 10.
Comparison of the LiPs electron density (blue) and positron density (red) calculated with the NEO-XCHF (solid lines) and SVM^{29} (dashed lines) approaches. Square roots of the densities have been plotted for more detailed representation of the valence electron portion of the electron density.
Comparison of the LiPs electron density (blue) and positron density (red) calculated with the NEO-XCHF (solid lines) and SVM^{29} (dashed lines) approaches. Square roots of the densities have been plotted for more detailed representation of the valence electron portion of the electron density.
Comparison of the LiPs (a) electron-positron contact density (black) and (b) electron density (blue) and positron density (red) calculated with the NEO-XCHF (solid lines) and NEO-HF (dashed lines) methods. The contact densities have been normalized. Square roots of the electron and positron densities have been plotted for more detailed representation of the valence electron portion of the electron densities.
Comparison of the LiPs (a) electron-positron contact density (black) and (b) electron density (blue) and positron density (red) calculated with the NEO-XCHF (solid lines) and NEO-HF (dashed lines) methods. The contact densities have been normalized. Square roots of the electron and positron densities have been plotted for more detailed representation of the valence electron portion of the electron densities.
Electron density for e^{+}LiH along the Li—H axis calculated with the NEO-XCHF (solid blue line) and ECG^{25} (dashed blue line) methods. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei. Note that the solid and dashed lines are virtually indistinguishable everywhere except at the Li and H nuclei.
Electron density for e^{+}LiH along the Li—H axis calculated with the NEO-XCHF (solid blue line) and ECG^{25} (dashed blue line) methods. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei. Note that the solid and dashed lines are virtually indistinguishable everywhere except at the Li and H nuclei.
Positron density for e^{+}LiH along the Li—H axis calculated with the NEO-XCHF and NEO-HF (solid lines) and ECG^{25} and HF^{11} methods by Strasburger (dashed lines). The NEO-XCHF and ECG curves are blue, while the HF-level curves are red. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei. Blue tick marks on the x axis represent the positions of the positron basis function centers in the NEO-XCHF calculations for this system.
Positron density for e^{+}LiH along the Li—H axis calculated with the NEO-XCHF and NEO-HF (solid lines) and ECG^{25} and HF^{11} methods by Strasburger (dashed lines). The NEO-XCHF and ECG curves are blue, while the HF-level curves are red. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei. Blue tick marks on the x axis represent the positions of the positron basis function centers in the NEO-XCHF calculations for this system.
Electron-positron contact density for e^{+}LiH along the Li—H axis calculated with the NEO (solid lines) and ECG^{25} (dashed line) methods. The NEO contact densities were calculated at the NEO-HF level and at the NEO-XCHF level with N _{gem} = 2 and N _{gem} = 8. The magnitudes of the maxima increase as the number of GTG functions increases. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei.
Electron-positron contact density for e^{+}LiH along the Li—H axis calculated with the NEO (solid lines) and ECG^{25} (dashed line) methods. The NEO contact densities were calculated at the NEO-HF level and at the NEO-XCHF level with N _{gem} = 2 and N _{gem} = 8. The magnitudes of the maxima increase as the number of GTG functions increases. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei.
Comparison of electron-positron contact densities for e^{+}LiH along the Li—H axis calculated with the ECG method^{25} (black dashed line) and at the HF level^{11} (blue solid line) by Strasburger, as well as with the NEO-HF method (red solid line). The HF-level contact densities have been scaled by 24 (Strasburger) and 20 (NEO-HF) and are very similar on this plot. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei.
Comparison of electron-positron contact densities for e^{+}LiH along the Li—H axis calculated with the ECG method^{25} (black dashed line) and at the HF level^{11} (blue solid line) by Strasburger, as well as with the NEO-HF method (red solid line). The HF-level contact densities have been scaled by 24 (Strasburger) and 20 (NEO-HF) and are very similar on this plot. The origin of the x axis is at the midpoint between the Li and H nuclei, and the circles indicate the positions of the Li (left circle) and H (right circle) nuclei.
Tables
Total energies (au), annihilation rates (ns^{−1}), and GTG function parameters, with γ given in units of au, calculated for PsH using the even-tempered 6s/6s positronic basis set. For each separate calculation, the energy and annihilation rate are given in the row corresponding to N _{gem}, the number of terms used in the GTG expansion (i.e., the energy and annihilation rate in a given row are associated with the column for which this row presents the last set of GTG function parameters).
Total energies (au), annihilation rates (ns^{−1}), and GTG function parameters, with γ given in units of au, calculated for PsH using the even-tempered 6s/6s positronic basis set. For each separate calculation, the energy and annihilation rate are given in the row corresponding to N _{gem}, the number of terms used in the GTG expansion (i.e., the energy and annihilation rate in a given row are associated with the column for which this row presents the last set of GTG function parameters).
Annihilation rates (ns^{−1}), radial expectation values (au), and average electron-electron, electron-positron, and electron-electron-positron contact densities (au) calculated for PsH with the NEO-HF, NEO-XCHF, SVM, and ECG approaches. The ⟨δ_{ ep }⟩ values for SVM and ECG have been multiplied by a factor of 2 to facilitate comparison of spin-projected quantities to NEO expectation values.
Annihilation rates (ns^{−1}), radial expectation values (au), and average electron-electron, electron-positron, and electron-electron-positron contact densities (au) calculated for PsH with the NEO-HF, NEO-XCHF, SVM, and ECG approaches. The ⟨δ_{ ep }⟩ values for SVM and ECG have been multiplied by a factor of 2 to facilitate comparison of spin-projected quantities to NEO expectation values.
Annihilation rates (ns^{−1}) for PsH, LiPs, and e^{+}LiH calculated with the NEO-HF, NEO-XCHF, SVM, and ECG approaches. In order to provide consistent comparisons, the constants used to calculate NEO annihilation rates with Eq. (9) were obtained from the corresponding SVM and ECG references for each system. These specific values for in Eq. (9) used to calculate the NEO annihilation rates are listed below after the corresponding reference in units of .
Annihilation rates (ns^{−1}) for PsH, LiPs, and e^{+}LiH calculated with the NEO-HF, NEO-XCHF, SVM, and ECG approaches. In order to provide consistent comparisons, the constants used to calculate NEO annihilation rates with Eq. (9) were obtained from the corresponding SVM and ECG references for each system. These specific values for in Eq. (9) used to calculate the NEO annihilation rates are listed below after the corresponding reference in units of .
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