^{1,a)}, P. López Ríos

^{1}and R. J. Needs

^{1}

### Abstract

Quantum Monte Carlo calculations of the first-row atoms Li–Ne and their singly positively charged ions are reported. Multideterminant-Jastrow-backflow trial wave functions are used which recover more than 98% of the correlation energy at the variational Monte Carlo level and more than 99% of the correlation energy at the diffusion Monte Carlo level for both the atoms and ions. We obtain the first ionization potentials to chemical accuracy. We also report scalar relativistic corrections to the energies, mass-polarization terms, and one- and two-electron expectation values.

We acknowledge financial support from the Cambridge Commonwealth Trust and UK Engineering and Physical Sciences Research Council (EPSRC). The calculations were performed on the Cambridge High Performance Computing Service.

I. INTRODUCTION

II. TRIAL WAVE FUNCTIONS

III. OPTIMIZATION

IV. RESULTS AND DISCUSSION

A. Atomic and ionic energies

B. Ionization potentials

C. Other expectation values

V. CONCLUSIONS

### Key Topics

- Wave functions
- 33.0
- Density functional theory
- 5.0
- Excitation energies
- 5.0
- Excited states
- 5.0
- Ground states
- 4.0

## Figures

Percentages of the correlation energy (% CE) retrieved for single-determinant Slater-Jastrow (SJ) and Slater-Jastrow-backflow (SJBF) wave functions using mean absolute deviation minimization (MADmin) and energy minimization (Emin).

Percentages of the correlation energy (% CE) retrieved for single-determinant Slater-Jastrow (SJ) and Slater-Jastrow-backflow (SJBF) wave functions using mean absolute deviation minimization (MADmin) and energy minimization (Emin).

Percentages of the correlation energy (% CE) retrieved for each atom within VMC and DMC. Chemical accuracy is achieved for Li–N and Ne at the DMC level.

Percentages of the correlation energy (% CE) retrieved for each atom within VMC and DMC. Chemical accuracy is achieved for Li–N and Ne at the DMC level.

Percentages of the correlation energy (% CE) retrieved for each ion within VMC and DMC. Chemical accuracy is achieved for Li^{+}–O^{+} at the DMC level. The values for F^{+} and Ne^{+} are within statistical uncertainty of chemical accuracy.

Percentages of the correlation energy (% CE) retrieved for each ion within VMC and DMC. Chemical accuracy is achieved for Li^{+}–O^{+} at the DMC level. The values for F^{+} and Ne^{+} are within statistical uncertainty of chemical accuracy.

Errors in the ionization potentials (Δ = IP_{calc} −IP_{ref}) for the first-row atoms obtained at the VMC and DMC levels compared to those from FCI-QMC, CCSD excitation, and CCSD-F12-HLC. The reference values are taken from Ref. 19. The shaded region represents chemical accuracy.

Errors in the ionization potentials (Δ = IP_{calc} −IP_{ref}) for the first-row atoms obtained at the VMC and DMC levels compared to those from FCI-QMC, CCSD excitation, and CCSD-F12-HLC. The reference values are taken from Ref. 19. The shaded region represents chemical accuracy.

## Tables

VMC variances for single-determinant (SD) and multideterminant (MD) Slater-Jastrow (SJ) and Slater-Jastrow-backflow (SJBF) wave functions. All variances are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

VMC variances for single-determinant (SD) and multideterminant (MD) Slater-Jastrow (SJ) and Slater-Jastrow-backflow (SJBF) wave functions. All variances are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

VMC and DMC energies of the first-row atoms and ions. Also included are Hartree–Fock energies *E* _{HF} calculated using ATSP2K (Ref. 5), the reference energies *E* _{ref} (Refs. 18 and 19), the correlation energies , and the percentage of the correlation energy recovered at the VMC level (VMC-corr%) and DMC level (DMC-corr%). All energies are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

VMC and DMC energies of the first-row atoms and ions. Also included are Hartree–Fock energies *E* _{HF} calculated using ATSP2K (Ref. 5), the reference energies *E* _{ref} (Refs. 18 and 19), the correlation energies , and the percentage of the correlation energy recovered at the VMC level (VMC-corr%) and DMC level (DMC-corr%). All energies are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

Comparison of the mean deviation (), mean absolute deviation (), and maximum deviation (Δ_{max}) of the ionization potentials obtained from several electronic structure methods. Deviations are from the reference nonrelativistic, clamped point nucleus values of Ref. 19. Averages were taken over Li–Ne, unless otherwise indicated. All values are in electron volts and the numbers in parentheses indicate the statistical uncertainty, if present, in the last digit shown.

Comparison of the mean deviation (), mean absolute deviation (), and maximum deviation (Δ_{max}) of the ionization potentials obtained from several electronic structure methods. Deviations are from the reference nonrelativistic, clamped point nucleus values of Ref. 19. Averages were taken over Li–Ne, unless otherwise indicated. All values are in electron volts and the numbers in parentheses indicate the statistical uncertainty, if present, in the last digit shown.

Scalar relativistic terms: mass-velocity (MV), electron–nucleus Darwin (D1), two-electron Darwin (D2), spin–spin contact interaction (SSC), retardation (Ret), and mass-polarization (MP) energies calculated at the DMC level. Values from the literature are given for Li, Li^{+}, Be, and Be^{+}. All values are in atomic units and the numbers in parentheses give the statistical uncertainty in the last digit shown.

Scalar relativistic terms: mass-velocity (MV), electron–nucleus Darwin (D1), two-electron Darwin (D2), spin–spin contact interaction (SSC), retardation (Ret), and mass-polarization (MP) energies calculated at the DMC level. Values from the literature are given for Li, Li^{+}, Be, and Be^{+}. All values are in atomic units and the numbers in parentheses give the statistical uncertainty in the last digit shown.

One-electron expectation values: electron moments for −2 ⩽ *n* ⩽ 3 and electron density at the coalescence point 〈δ(*r* _{ i })〉, summed over all electrons *i*. All values are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

One-electron expectation values: electron moments for −2 ⩽ *n* ⩽ 3 and electron density at the coalescence point 〈δ(*r* _{ i })〉, summed over all electrons *i*. All values are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

Two-electron expectation values: interelectronic moments for −2 ⩽ *n* ⩽ 3, the electron-pair density at the coalescence point 〈δ(*r* _{ ij })〉, and the mass-polarization term , summed over all electron pairs *ij*. All values are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

Two-electron expectation values: interelectronic moments for −2 ⩽ *n* ⩽ 3, the electron-pair density at the coalescence point 〈δ(*r* _{ ij })〉, and the mass-polarization term , summed over all electron pairs *ij*. All values are in atomic units and the numbers in parentheses indicate the statistical uncertainty in the last digit shown.

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