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Assessing weak hydrogen binding on Ca+ centers: An accurate many-body study with large basis sets
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Image of FIG. 1.
FIG. 1.

AFQMC total energy error, E mCD(δ) − E DD, as a function of the mCD threshold parameter δ, where E DD is the energy obtained from direct diagonalization of the Coulomb matrix (energies are given in Table I). The width of the gray line indicates the statistical uncertainty of E DD. The corresponding error (with respect to the UHF energy) of the trial wave function variational energy is also shown.

Image of FIG. 2.
FIG. 2.

Log-log plots of wall clock times (seconds) vs. basis size M, comparing DD () and mCD (▲) of the ERI supermatrix V μν. The DD is carried out with 8-thread OPENMP (see text). The mCD timing is obtained from a locally modified MPQC (Ref. 45) program, for fixed δ = 10−6. No multithreading is used in the mCD calculations. The dashed (dashed-dotted) lines are linear regressions. The DD slope ∼M 5.7 is consistent with the expected M 6 scaling, while mCD scales as ∼M 3.1.

Image of FIG. 3.
FIG. 3.

An illustration of the Ca+– 4H2 model chemistry, containing one Ca+ surrounded by four hydrogen molecules in a D 4h symmetric configuration.

Image of FIG. 4.
FIG. 4.

All-electron GTO-AFQMC and MP2 PEC of the Ca+– 4H2 symmetric dissociation as a function of Z (Ca+– H2 separation distance) at the cc-pCVTZ basis level. The solid lines are from separate Morse fits for the inner and outer regions. Also shown at selected Z are GTO-AFQMC results from larger cc-pCVQZ and cc-pCV5Z basis sets. Vertical lines at Z = 2.3 Å and Z = 4.0 Å are a guide to the eye.

Image of FIG. 5.
FIG. 5.

Basis set convergence of GTO-AFQMC Ca+– 4H2 total energies for Z = 2.3 Å (near the inner well minimum). Energies are plotted as a function of x −3, where x ∈ {3, 4, 5} is the correlation consistent basis cardinal number. Left panel: valence-only cc-pVxZ and aug-cc-pVxZ; right panel: core-valence cc-pCVxZ and aug-cc-pCVxZ.

Image of FIG. 6.
FIG. 6.

Top panels: The binding energy of Ca+– 4H2 system near the inner well minimum (Z = 2.3 Å, d H − H = 0.7682 Å) plotted against (x −3), where x is the correlation consistent basis cardinal number. Bottom panels: The binding energy near the outer well minimum (Z = 4.0 Å, d H − H = 0.7362 Å).

Image of FIG. 7.
FIG. 7.

The PEC of the Ca+– 4H2 symmetric dissociation extrapolated to the CBS limit. The PEC is shown as a function of Ca+– H2 separation distance, Z. Morse fits to the AFQMC points are shown as solid curves. The gray shading provides an estimate of the PEC uncertainties. Dotted lines shows the continuation of the outer well PEC into the small Z region. Morse fits to the MP2 points are shown as dashed curves. The extrapolation scheme is discussed in Sec. II C. For comparison, binding energies computed using very large aug-cc-pCV5Z basis sets are also shown at Z = 2.3 and 4.0 Å.


Generic image for table
Table I.

GTO-AFQMC total energies for several values of the mCD threshold parameter δ for Ca+– 4H2 (Z = 2.3 Å and d H − H = 0.7682 Å; see text), using the cc-pVTZ basis (M = 155). A fixed Trotter time step was used for all the calculations. The total energy E DD, obtained from direct diagonalization of V μν, is presented for comparison; the eigenvalue cutoff is also shown. N γ is the corresponding number of auxiliary fields. A full rank, symmetric, positive definite V μν matrix would have required 1552 = 24 025 Cholesky vectors. All energies are in E h.

Generic image for table
Table II.

All electron GTO-AFQMC Ca+– 4H2 total energies (in E h) for two geometries and two correlation-consistent core-valence basis sets. “Inner well” corresponds to Z = 2.3 Å and d H − H = 0.7682 Å; “outer well” corresponds to Z = 4.0 Å and d H − H = 0.7362 Å. The total energies of the isolated Ca+ and H2 fragments are also shown. The mCD method with δ = 10−6E h is used, unless otherwise noted. In all cases, bias from mCD is much smaller than the AFQMC statistical error, which are on the last two digits and are indicated in parentheses. All energies were extrapolated to the τ → 0 limit. CBS indicates the complete basis set extrapolated limit.

Generic image for table
Table III.

Binding energies E b after extrapolation to the CBS limit. The two geometries are the same as in Table II (inner well at Z = 2.3 Å and outer well at Z = 4.0 Å). The contributions to the AFQMC binding energy [E b(Z, ∞) on the third row] from HF and correlation are shown separately in the first two rows. Energies are in eV. AFQMC statistical errors are shown in parentheses.

Generic image for table
Table IV.

Spectroscopic constants associated with the AFQMC inner and outer wells depicted in Fig. 7. These parameters were obtained by fitting a Morse curve to the two wells separately. The error bars shown in parentheses below include both the fitting and AFQMC statistical uncertainties.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Assessing weak hydrogen binding on Ca+ centers: An accurate many-body study with large basis sets