2025 Volume 11 Article ID: 2024-0032
Multiple-precision Hartree–Fock equations are solved for the ground states of the He, Li, and Be atoms and highly accurate wave functions are determined. The numbers of significant figures of the total energies are 66 for He, 43 for Li, and 43 for Be. The Lambda function, which is a Laguerre-type basis and can be used to construct a complete orthonormal system, acts as the basis function. Using the resulting Hartree–Fock wave functions, nucleus-electron cusp conditions are also calculated. Their significant figures are 30 for He, 19 for Li, and 19 for Be, in close proportion to the numbers of significant figures for the total energies. In fact, the ratios are 0.45, 0.44, and 0.44, respectively. The radial expectation values, < ri > (i = −2, −1, 1, ... 9), are also investigated. It is verified that the computational accuracy varies depending on the physical property.
The purpose of this study is to accurately solve the atomic Hartree–Fock (HF) equation and compare the accuracy of the calculated physical properties, including ρ(0) (electron density at the nucleus), CC (nucleus-electron cusp condition) [1,2,3], and radial moments (< ri > , i = −2, −1, 1, ... 9), with the accuracy of the total energy (TE). First, we will give an overview of our previous work.
To date, accurate atomic HF solutions have been obtained mainly by numerical Hartree–Fock (NHF) method [4,5,6,7,8,9]. Efforts have also been made to develop more accurate solutions [10, 11]. Recently, Cinal [12] applied the pseudospectral method to solve the closed-shell ground states of 18 atoms and obtained HF TEs with 25 significant digits. Other solution methods include the piecewise polynomial [13], the B-spline expansion [14], and the basis set expansion [15,16,17].
We used the Λ function [18,19,20,21], which is a Laguerre-type basis function [22], to solve the HF equation. The basis function has the following form:
| (1) |
| (2) |
Here, for the radial function Rnl, we use the Λ function. The
term Ylm denotes the spherical harmonics, n is
the principal quantum number,
The Λ function is used to construct a complete orthonormal system for an atomic system. Due to the orthogonality, the overlap matrix in the Hartree–Fock–Roothaan (HFR) equation [24] is a unit matrix, and the set is free from linear dependence issues [25]. This allows us to easily extend the scale of the basis functions. Since the set is complete for bound systems, it is guaranteed that in the nonrelativistic domain, the HF TE converges monotonically from above to the HF limit as the number of expansion terms (N) is increased. The atomic system is a one-center system, and the one- and two-electron integrals [26, 27] can be calculated analytically and precisely Therefore, the Λ function is suitable for solving the atomic HF equation accurately. In addition, since the Λ function is an exponential-type orbital (ETO), it is also suitable for calculating CC.
We performed HF calculations [28,29,30,31,32] on atoms, using the Λ function as the basis function. In these calculations, the maximum numbers of basis functions used were 150, 149, 148, and 147 for the s, p, d, and f symmetries, respectively. For brevity, we designate this expansion as N = 150. All the calculations were performed with quadruple precision. The integrals were calculated to have 30 significant decimal digits, and the HF TE of He was obtained to 30 significant digits. This is the largest number to date. Since the machine epsilon in IEEE quadruple precision, which uses 16 bytes for one real number, is 1.9 × 10−34, the number of significant figures (NSF) in quadruple precision cannot exceed 34.
In our recent paper [32], we investigated the accuracy of total HF energy and various other properties using accurate HF solutions. This gave the ratio of the NSF of the calculated property to the NSF of the TE. We verified that this ratio is nearly constant, regardless of the atomic species.
The NSF of HF TE is obtained by comparing the value at N = 150 with the limit value of N → ∞. However, since the exact value of CC is known [1], its NSF can be calculated without considering the limit. Therefore, among the one-electron properties, CC is an excellent test example for calculating the ratio of significant figures. We then performed HF calculations using the Λ function for seven elements in Group 18 (i.e., He, Ne, Ar, Kr, Xe, Rn, and Og) and seven elements from Li to F in the second row of the periodic table. The NSFs of the HF TEs were 30, 30, 30, 23, 20, 17, and 15 for He, Ne, Ar, Kr, Xe, Rn, and Og, respectively, and the NSFs of the CCs were 26, 16, 14, 9, 9, 6, and 6, respectively. The corresponding ratios were 0.87, 0.53, 0.47, 0.39, 0.45, 0.35, and 0.40, respectively. For Ar–Og, the ratio was about 0.4, while it was larger for He and Ne.
These results occurred because the accuracy of the integrals is limited to 30 orders of magnitude in quadruple precision. Therefore, the NSF for the TE is limited to 30. That is, the HF TE at N = 150 reached the limit of quadruple precision in the calculations of the atoms in the first and second rows. We also found that the calculated HF TEs for He, Li, and Be achieved 30-digit accuracy at N = 52, N = 94, and N = 84, respectively. Calculations beyond quadruple precision are required to obtain the correct ratio for these atoms.
We estimated the NSF ratios if multiple-precision arithmetic was used, as follows. For Ar, Kr, Xe, Rn, and Og, the quadruple-precision limit for TE was not reached. Therefore, the values of the NSFs for these five atoms are valid, even when multiple-precision arithmetic is used. By extrapolating the NSFs of Og, Rn, Xe, Kr, and Ar to the light atoms, we estimated the NSFs of the TEs for the light atoms, which were not affected by rounding error. We then predicted 53 for He, 48 for Li, 44 for Be, and 34 for Ne. This is where we reached in our preceding paper [32].
Let us now turn to the present study. This time we will study the ground states of the He, Li, and Be atoms. Due to the high computational cost of p electrons, we will only consider these three atoms with s electrons (l = 0).
To solve the HF equation with high accuracy, two conditions are required:
(i) Many expansion terms.
(ii) A large number of bits for arithmetic operations.
The first condition reduces the truncation error, and the second condition reduces the rounding error. It is desirable for both errors to be of a similar order of magnitude. To obtain TE close to the HF limit, the number of expansion terms must be increased to reduce the truncation error. An expansion of 150 terms is sufficient for this purpose. And the rounding error must be reduced to less than the truncation error. We will reduce the rounding error and increase the NSF by changing the arithmetic from quadruple precision to multiple precision. Using multiple-precision arithmetic [33, 34], it is possible to overcome the quadruple precision limit and correctly calculate the ratios of the NSFs for He, Li, and Be. Throughout this paper we use atomic units (au).
We used Bailey’s arbitrary precision package MPFUN90 [33] for the multiple-precision calculations; its latest version is MPFUN2020 [34]. In MPFUN90, the number of bits used to represent a real number is assigned by a constant, mpipl, and the precision in a real computation is specified by a parameter, init. From the quadruple-precision HF calculations [32] for Ar, Kr, Xe, Rn, and Og, we estimated that the NSF of TE of He at N = 150 would be 53 if the multiple-precision calculation was performed. In reality, during our multiple-precision calculation, we expected that the number of significant figures would exceed 53, and would reach about 65. Accordingly, we found it necessary to specify the value of init so that the integrals should have 80 significant digits and the final rounding error for TE should be less than 1.0E-70. As detailed in the next section, we set mpipl to 350 and also set init to 350. When mpipl = 350, one multiple-precision variable occupies 216 bytes ( = 54 words) of memory.
2.2 Verification of the accuracy of integral valuesThe algorithm for calculating integrals is the same as before [32]. We first calculated kinetic energy integrals according to the analytical formulation of Hagstrom and Shull [26]. Second, we calculated electron-nucleus interaction integrals and two-electron integrals according to the formulation of Freund and Hill (abbreviated as FH) [27], which is based on a recursion formula. In the FH method, especially when N is large, it is necessary to use multiple-precision arithmetic to prevent the loss of significant digits during the calculation. We explored the value of init to ensure sufficient accuracy. We also verified the accuracy of the integral values calculated by the FH method by comparing them with the values evaluated by numerical integration using the Gauss–Laguerre formula (abbreviated as GLa) [31].
Table 1 shows some electron-nucleus attraction integrals. As n increases, the Λ function has more nodes and oscillates more violently, so that the accuracy of the calculated integrals must be checked. Table 1 lists some of the integral values up to 80 digits evaluated at init = 370. Since the value at init = 350 and the value at init = 370 match up to 80 digits, we regard the value at init = 370 as the true value. The err (init50) and err (init350) columns of Table 1 list the difference between the integral values at init(50 and 350) and the value at init = 370. The integral value at init = 350 is identical to the integral value at init = 370 up to 80 digits. The column err (GLa) shows the difference between the integral value evaluated by the numerical integration using the GLa method and that by the FH method at init = 370. Up to 79 digits, both integrals are identical.
| n | n’ | Value of (n | 1/r | n’) evaluated with init=370 | err(init50) a) | err(init350) b) | err(GLa) c) |
| 149 | 150 | –9.93355 40814 32371 29546 35139 1251103849 13748 40474 83900 61253 1144389349 98834 59285 9333E-1 | -6.13E-52 | EQ80 | 3.03E-81 |
| 150 | 151 | –9.93399 26779 87828 54899 56379 0387646981 34502 56190 29392 12346 0345645488 25288 86733 9621E-1 | -1.24E-52 | EQ80 | 7.16E-81 |
a) Difference between the integral value evaluated at init = 50 and that at init = 370.
b) Difference between the integral value evaluated at init = 350 and that at init = 370. "EQ80" indicates that both values are identical up to 80 digits.
c) Difference between the integral value evaluated with the Gauss–Laguerre quadrature at init = 350 and that evaluated with the FH method at init = 370.
Table 2 shows some two-electron integrals. The radial components (Fl) are compared. Differences between the values of Fl evaluated at init(290, 330, and 350) and the value at init = 370 are listed in the columns labeled err (init290), err (init330) and err (init350). From Table 2 it is clear that the integrals at init = 350 have an accuracy of at least 80 digits.
| n | Value of Fl evaluated with init=370 | err(init290) a) | err(init330) b) | err(init350) c) | err(GLa) d) |
| 149 | 1.81375 55660 68543 05049 62166 3938936174 62958 84303 24913 76446 4935595160 94367 26203 9528E-2 | -2.41E-37 | 1.64E-72 | EQ80 | 1.29E-81 |
| 150 | 1.80349 90496 12097 47442 51703 3562132013 03450 04137 47076 74973 7000922534 02803 22090 9052E-2 | 6.74E-37 | 1.74E-73 | EQ80 | -2.05E-81 |
a) Difference between the integral value evaluated at init = 290 and that at init = 370.
b) Difference between the integral value evaluated at init = 330 and that at init = 370.
c) Difference between the integral value evaluated at init = 350 and that at init = 370. "EQ80" indicates that both values are identical up to 80 digits.
d) Difference between the integral evaluated with the Gauss–Laguerre quadrature at init = 350 and that evaluated with the FH method at init = 370.
The err (GLa) column in Table 2 shows the difference between the value evaluated by the GLa method based on Moseley’s formulation [35, 31] and the value from the FH method. For this calculation we converted our previously developed Gauss–Laguerre quadrature program [31] from quadruple precision to multiple precision. We obtained the zeros of the Laguerre polynomial Ln(x) and the corresponding weights, both of which are necessary for the quadrature, to 80 digits of precision using Wolfram Mathematica [36], following the procedure of Golub and Welsch [37]. Table 2 shows that the integral evaluated by the GLa method at init = 350 and that by the FH method at init = 370 are identical up to 78 digits. The two integrals calculated independently by the different methods are in good agreement.
Since the integrals calculated according to the analytical formula at init = 350 are sufficiently accurate as described above, we saved these integrals in a file, and we also use init = 350 in the subsequent SCF calculations. We saved the integrals for the nuclear charge Z = 1 and the exponent ζ = 1 to a file. When solving the SCF equation for each atom, the stored integrals are used after being multiplied by Z and ζ.
2.3 SCF convergence criterionFor the SCF calculations, we used the atmscf program, which we converted to multiple precision from the original double-precision version written by Pitzer [38]. The atmscf program solves the eigenvalue problem of the SCF equation by the method proposed by Roothaan and Bagus [24]. Convergence of the SCF is judged by whether the largest absolute differences in the AO coefficients in successive iterations fall below a threshold. This threshold is given by the variable named threl in the program.
It is natural to expect the tighter threshold (threl) to give a more accurate wave function. To explore the most appropriate value for threl, we examined the rate of change of TE (ΔTE) by varying threl. We chose the He wave function at N = 150 for ζ = 5.1448 for this test. As detailed in the next section, TE has the lowest value at this value of ζ. We believe that the wave function obtained with this ζ is close to the exact HF solution. We use the TE value at threl = 1.4E-70 as the reference value, and express it as TE (B). We also examine the rate of change ΔVR of the virial ratio (VR = <V> / <T>) [39]. Since an accurate wave function gives (−2) for VR, we set (−2) as the reference value of VR (B).
ΔTE = (TE − TE(B))/| TE(B) | ΔVR = (VR − VR(B))/| VR(B) |Here, we treat ΔTE and ΔVR as functions of threl.
We reduced threl as 1.2E-32, 1.1E-41, 1.0E-50, 1.6E-55, 1.5E-64, 1.4E-70, 1.4E-73. At threl = 1.2E-32, ΔTE was 2.76E-64 but, at the other values of threl, TEs were identical to TE (B) up to 80 digits. At the same time, ΔVR varied as 5.1E-33, 4.5E-43, 6.5E-52, −1.6E-56, −2.1E-65, 0.0E00, −2.1E-71. Thus the absolute values of ΔVR decreased rapidly. Let us measure the accuracy of the wave function by VR. At threl = 1.4E-70, VR matched (−2) up to 69 digits. Indeed, the NSF of VR was as great as the NSF of TE. We may therefore say that an accurate wave function can be obtained at threl = 1.4E-70. We chose to specify 1.4E-70 for threl in the following SCF calculations.
2.4 Optimization of ζAt the given number of expansion terms, N, we minimized HF TE by varying ζ. For He, Li, and Be at N = 150, Table 3 shows ΔTE (ζ) (the rate of variation of TE with respect to ζ) and the change of the virial ratio, VR (ζ),
ΔTE(ζ) = [ TE(ζ) − TE(ζopt) ]/| TE(ζopt) | (3) δVR(ζ) = VR(ζ) − (−2) = VR(ζ) + 2 (4)| He | Li | Be | ||||||||
| ζ | ΔTE (ζ) | δVR | ζ | ΔTE (ζ) | δVR | ζ | ΔTE (ζ) | δVR | ||
| 5.1440 | 3.84E-71 | -5.00E-67 | 3.424534308209212 | 4.11E-71 | -2.13E-55 | 4.389148271030510 | 1.23E-69 | -1.27E-54 | ||
| 5.1447 | 4.95E-73 | -5.84E-68 | 3.424534308209213 | ZERO | -6.88E-56 | 4.389148271030515 | 2.04E-70 | -5.24E-55 | ||
| 5.1448 | ZERO | +6.26E-69 | 3.424534308209214 | 9.83E-73 | +7.53E-56 | 4.389148271030518 | ZERO | -7.38E-56 | ||
| 5.1449 | 7.37E-73 | +6.96E-68 | 3.424534308209215 | 4.40E-71 | +2.19E-55 | 4.389148271030520 | 3.47E-71 | +2.26E-55 | ||
| 5.1450 | 2.71E-72 | +1.32E-67 | 3.42500000000000 | 4.57E-48 | +6.73E-44 | 4.389148271030540 | 7.90E-69 | +3.23E-54 | ||
a) ΔTE (ζ) denotes [ TE (ζ) − TE (ζopt) ] / | TE (ζopt) | . TE: total energy.
δVR denotes (VR + 2). VR: virial ratio.
"ZERO" in the ΔTE (ζ) column indicates TE evaluated with ζopt. ζ on this row is ζopt (written in bold).
In equation (3), ζopt signifies ζ that gives the lowest TE. Here, we treat ΔTE and δVR as functions of ζ.
Table 3 shows that ζopt is 5.1448 for He. This is written in bold in the row in which “ZERO” is typed in the ΔTE column. In the interval between ζ = 5.1440 and ζ = 5.1450, ΔTE is smaller than 1.0E-70. This implies that the TE values are identical to the lowest value up to at least 70 digits. Table 3 also shows that the absolute value of δVR is least (6.26E-69) at ζopt. The VR at ζ (< ζopt) is smaller than (−2) (i.e., δVR < 0), and VR at ζ (> ζopt) is greater than (−2) (i.e., δVR > 0). δVR changes more than ΔTE, indicating that δVR is much more sensitive to the accuracy of the wave function.
As written in the literature [39], “the fulfillment of the virial theorem is a necessary but not sufficient criterion that a wave function is an accurate solution of the Schrödinger equation”. Since the wave function we obtained here with ζopt is quite close to the exact HF solution which satisfies the virial theorem, it is acceptable to use VR to assess the accuracy of this wave function.
Table 3 shows also that ζopt is 3.424534308209213 for Li, and 4.389148271030518 for Be. For Li and Be, more precise values of ζ are required than for He, because their TEs vary more sensitively with ζ.
2.5 Error and number of significant figuresIn textbooks on numerical computation [40] the number of significant figures is defined as follows;
{absolute error} = {approximation} − {true value} (5) {relative error} = {absolute error}/{true value} (6) {number of significant figures} = − log10 (| {relative error} |) (7)We follow this definition in this work. Here, {approximation} refers to the value obtained by a numerical calculation. {number of significant figures} in equation (7) is a real number, which we call RNSF (real number of significant figures) below. We designate the integer obtained by rounding down RNSF as NSF.
Since the calculated value contains errors, it is usually almost impossible to obtain the true value. There are exceptions where the exact values are known, however. The nucleus-electron cusp condition (CC) [1,2,3] is one such example, with the exact value (–Z). CC is defined by the following formula:
| (8) |
Here, Z is the nuclear charge and ρ(0) is the electron density at the nucleus.
Since the Λ functions construct a complete orthonormal system, it follows that, for non-relativistic HF, TE converges monotonically to the HF-limit as N increases. We therefore calculate the NSF of TE by treating the limit value for N → ∞ as the {true value} in the sense of equations (5–6). Since TE varies as N increases, it is useful to determine the value of TE up to the digit that remains unchanged as N approaches infinity. We will report this value as the recommended value for TE. We determine the recommended value by rounding the calculated TE value at N = 150 to the number of digits specified by the NSF.
The limit value for N → ∞ is obtained by taking two points (Na and Nb) before N = 150, selecting several points between Na and Nb as sampling points, and fitting them to the following exponential function [32], then extrapolating the fitting curve to infinity.
TE(N) ~a exp(– k (N − Na)) + c (9)Here, a, k, and c are fitting parameters. This c is the limit value for N → ∞.
Unlike TE, expectation values such as < ri > are not guaranteed to converge monotonically. We calculate their NSFs by regarding the limit values for N → ∞ as the {true values} in the sense of equations (5–6).
To obtain the N → ∞ limit of TE, the exponential function in equation (9) is fitted as a model function and is extrapolated. In the case of the He atom, however, there is a point that must be discussed in detail. Figure 1 plots δTE (N) = [ TE (N) − TE (150) ] against N. The value of δTE (N) falls exponentially to zero, but like a staircase, because the decrease of the TE as N(odd) → N+1 is much smaller than the decrease as N → N+2.
[TE (N) − TE (150)] for He vs N.
We tried three ways to select sample points for the fitting.
● five sample points, N = 141, 143, 145, 147, 149. (odd)
● five sample points, N = 142, 144, 146, 148, 150. (even)
● ten sample points, N = 141–150. (all)
Figure 2 shows the results. Extrapolation of δTE (N) to large N(> 150) shows that the two fitting curves, one (odd) with the odd-numbered sample points and the other (even) with the even-numbered sample points, are close to each other and converge to almost the same value. We used the average of the limit values from (odd) and (even) as TE (∞). If, however, we extrapolate the fitting curve (all) using all the odd and even sample points, its limit value will deviate from the average. This energy difference is 2.9E-65. For He, we also do the same odd and even fittings for the expectation value <A> as for the TE, and use their average as <A(∞)>. For Li or Be, TE (and also <A> ) changes smoothly with N and converges to a single value, so the averaging is unnecessary.
Fitting curves for [TE (N) − TE (∞, all)] for He. The blue curve (odd) shows the case where sample points are taken for the odd-numbered N, and the red curve (even) does the same for even-numbered N; the black dashed curve (all) is the case where all sample points are taken. TE (∞, all) is the total energy obtained by extrapolating the fitting curve obtained with all of the sample points to infinity.
We calculate the RNSF of TE (N) (or <A>(N)) using the following formula:
RNSF(TE(N)) = − log10 [ | TE(N) − TE(∞) |/| TE(∞) | ] (10)Figure 3 plots the RNSFs of TE and <A > for He against N = 50–150. The graphs of RNSFs extend almost linearly. In quadruple precision [32], the graph of the RNSF of TE stopped at 30. As shown in Figure 3, however, it extends up to 66. This tells us that the best performance of the Λ functions of N = 150 was not fully exploited in the previous study [32], due to rounding error in quadruple precision.
[– RNSF (< A > )] for He vs N.
For the He atom, ζ has been optimized until the variation of TE is smaller than 1.0E-70. Table 4 shows the results for He. Table 4 also lists literature values of TE. The TE value at N = 150 is listed in the fourth column of the second row. Since this value has 70 significant digits, it is folded so that it does not extend beyond the column. If N is increased, TE (N) will continue to decrease further. The limit value of N → ∞ obtained from equation (9) is listed in the third row. Substituting the TE (150) and the TE (∞) into equation (10) gives the RNSF, and then truncating the RNSF gives the NSF, which is listed in the second column. The recommended value, shown in the first row, is obtained by rounding down the TE (∞) to the nearest NSF digit. The RNSF of TE is 66.3, and the NSF is 66. In the previous quadruple-precision calculation [32], the NSF was 30, which has been greatly extended.
| Property a) | NSF b) (RNSF) | Ratio c) | Value | Ref. d) |
| TE | 66 (66.3) | 1.00 | –2.86167 99956 12238 87877 55437 40022 5634294128 79776 75244 70539 06864 69743 | Rec. |
| –2.86167 99956 12238 87877 55437 40022 5634294128 79776 75244 70539 06864 69743 3503 | N = 150 | |||
| –2.86167 99956 12238 87877 55437 40022 5634294128 79776 75244 70539 06864 69743 4806 | N → ∞ | |||
| 30 | –2.86167 99956 12238 87877 55437 4002 | [28] | ||
| 25 | –2.86167 99956 12238 87877 5544 | [12] | ||
| 14 | –2.86167 99956 122 | [13] | ||
| 13 | −2.86167 99956 12 | [17] | ||
| 13 | −2.86167 99956 12 | [16] | ||
| 12 | –2.86167 99956 1 | [14] | ||
| 10 | −2.86167 9996 | [10] | ||
| 10 | –2.86167 9996 | [7] | ||
| 10 | –2.86167 9996 | [6] | ||
| VR | 68 (68.5) | 1.03 | –2.00000 00000 00000 00000 00000 00000 0000000000 00000 00000 00000 00000 00000 00 | Rec. |
| CC | 30 (30.3) | 0.45 | –2.00000 00000 00000 00000 00000 0000 | Rec. |
| ρ(0) | 32 (32.6) | 0.48 | +3.59591 82637 62248 92070 69875 63001 7 | Rec. |
| r −2 | 35 (35.2) | 0.53 | +5.99550 36673 12524 59727 35587 42220 3366 | Rec. |
| r −1 | 65 (65.4) | 0.98 | +1.68728 22152 81009 40400 80496 39502 5437250446 25731 01112 90065 15424 2332 | Rec. |
| r | 63 (63.1) | 0.95 | +9.27273 40473 14931 97418 92770 72297 1598064427 00613 22365 80298 54524 99E-1 | Rec. |
| r 2 | 61 (61.6) | 0.92 | +1.18482 84799 09482 92632 57589 05407 3484255233 53308 21288 81335 43575 | Rec. |
| r 3 | 60 (60.2) | 0.91 | +1.94063 62073 75593 29252 71080 42338 7024142085 08018 61031 91798 1459 | Rec. |
| r 4 | 58 (58.8) | 0.88 | +3.88794 66255 72633 62739 15882 63622 6854816768 09787 65625 41375 39 | Rec. |
| r 5 | 57 (57.5) | 0.86 | +9.22119 73006 38698 13804 88253 04640 7209456243 59111 59249 50986 1 | Rec. |
| r 6 | 56 (56.2) | 0.85 | +2.52783 90413 29334 76415 89554 55045 8912261335 57154 01244 44909E1 | Rec. |
| r 7 | 54 (54.9) | 0.82 | +7.86524 91919 01012 08577 91959 89351 4150554211 96317 99910 758E1 | Rec. |
| r 8 | 53 (53.7) | 0.80 | +2.73845 06809 01483 72331 15087 38385 1353781261 85796 34911 40E2 | Rec. |
| r 9 | 52 (52.6) | 0.79 | +1.05483 26442 59784 46180 60475 93002 1437277364 02123 58509 3E3 | Rec. |
a) TE: total energy. VR: virial ratio.
b) NSF: Number of significant figures. RNSF: Real number of significant figures (written in parentheses).
c) Ratio of NSF to the NSF of TE.
d) "Rec." denotes the value recommended by the present study.
As in the case of TE, the recommended value of CC was obtained from the value of CC at N = 150 and the limit value of N → ∞. To save space, only the recommended value is shown in Table 4. The NSF of CC is 30. Our previous quadruple-precision calculation [32] yielded 26 for the NSF, which has also been extended. The ratio of the NSF of CC to the NSF of TE is 0.45 ( = 30/66). This ratio is shown in the third column of Table 4. This is close to the value of 0.43 estimated from the quadruple-precision calculation in the previous study [32], indicating that the estimation method was appropriate.
Table 4 also lists the virial ratio, the electron density at the nucleus (ρ(0)) and the radial expectation values. As seen from the third column, the NSFs of the radial expectations are smaller than the NSF of the HF total energy.
3.3 The lithium atomTable 5 shows results for the Li atom. The NSF of TE is 43 in the present multiple-precision calculation, whereas its estimated value [32] was 48. The NSF of CC is 19, and the ratio of the NSF of CC to the NSF of TE is 0.44. This value is almost equal to that of 0.43 for He.
| Property a) | NSF b) (RNSF) | Ratio c) | Value | Ref. d) |
| TE | 43 (43.2) | 1.00 | –7.43272 69307 29757 84257 90238 56145 8376910681 98 | Rec. |
| –7.43272 69307 29757 84257 90238 56145 8376910681 98481 77252 05750 93388 54657 3445 | N = 150 | |||
| -7.43272 69307 29757 84257 90238 56145 8376910681 98982 22347 38021 22283 44660 9217 | N → ∞ | |||
| 30 | -7.43272 69307 29757 84257 90238 5615 | [30] | ||
| 12 | –7.43272 69307 3 | [14] | ||
| 10 | –7.43272 6931 | [6] | ||
| VR | 55 (55.5) | 1.28 | -2.00000 00000 00000 00000 00000 00000 0000000000 00000 00000 0000 | Rec. |
| CC | 19 (19.0) | 0.44 | –3.00000 00000 00000 000 | Rec. |
| ρ(0) | 21 (21.0) | 0.49 | +1.38148 19872 06468 49620E1 | Rec. |
| r −2 | 23 (23.3) | 0.53 | +1.00706 42535 99489 27313 07 | Rec. |
| r −1 | 41 (41.8) | 0.95 | +1.90515 27763 33801 65035 25209 26668 4974138147 | Rec. |
| r | 39 (39.4) | 0.91 | +1.67330 45304 28967 03898 69585 22401 63216140 | Rec. |
| r 2 | 38 (38.1) | 0.88 | +6.21068 91473 92334 09945 24992 26685 4037333 | Rec. |
| r 3 | 37 (37.0) | 0.86 | +3.15852 72670 31471 52124 65673 28568 173137E1 | Rec. |
| r 4 | 35 (35.9) | 0.81 | +1.88981 97727 08646 12039 19140 85396 9231E2 | Rec. |
| r 5 | 34 (34.8) | 0.79 | +1.27971 58590 15096 33990 37755 10126 863E3 | Rec. |
| r 6 | 33 (33.8) | 0.77 | +9.65439 34146 32057 20057 38167 25535 79E3 | Rec. |
| r 7 | 32 (32.8) | 0.74 | +8.03093 28341 86643 95493 42961 96828 2E4 | Rec. |
| r 8 | 31 (31.9) | 0.72 | +7.30548 04909 74822 16836 50687 68514E5 | Rec. |
| r 9 | 31 (31.0) | 0.72 | +7.21669 77327 49538 78567 95139 33334E6 | Rec. |
a) TE: total energy. VR: virial ratio.
b) NSF: Number of significant figures. RNSF: Real number of significant figures (written in parentheses).
c) Ratio of NSF to the NSF of TE.
d) "Rec." denotes the value recommended by the present study.
Table 6 shows results for the Be atom. The NSF of TE was 30 in the previous quadruple-precision calculation [32]. It is 43 in the present multiple-precision calculation, while its estimated value [32] was 44. This shows that the estimation in the previous paper was correct. The NSF of CC is 19, and the ratio of the NSF of CC to the NSF of TE is 0.44.
| Property a) | NSF b) (RNSF) | Ratio c) | Value | Ref.d) |
| TE | 43 (43.1) | 1.00 | –1.45730 23168 31639 95824 11447 35987 0014453877 72E1 | Rec. |
| -1.45730 23168 31639 95824 11447 35987 0014453877 71983 81295 97003 82653 80788 3580E1 | N = 150 | |||
| –1.45730 23168 31639 95824 11447 35987 0014453877 72109 83871 24723 00204 65754 5988E1 | N → ∞ | |||
| 30 | -1.45730 23168 31639 95824 11447 3599E1 | [30] | ||
| 25 | –1.45730 23168 31639 95824 1145E1 | [12] | ||
| 12 | –1.45730 23168 3E1 | [14] | ||
| 11 | −1.45730 23168E1 | [10] | ||
| 10 | –1.45730 2317E1 | [7] | ||
| 10 | –1.45730 2317E1 | [6] | ||
| VR | 55 (55.4) | 1.28 | –2.00000 00000 00000 00000 00000 00000 0000000000 00000 00000 0000 | Rec. |
| CC | 19 (19.0) | 0.44 | –4.00000 00000 00000 000 | Rec. |
| ρ(0) | 20 (20.9) | 0.47 | +3.53877 16741 09378 6449E1 | Rec. |
| r −2 | 23 (23.2) | 0.53 | +1.44045 06445 44157 38993 24E1 | Rec. |
| r −1 | 41 (41.6) | 0.95 | +2.10219 94129 30678 79400 72225 48834 5576562341 | Rec. |
| r | 39 (39.4) | 0.91 | +1.53220 39722 81112 53930 21732 84545 63027792 | Rec. |
| r 2 | 38 (38.1) | 0.88 | +4.32969 28432 10096 87398 12163 00197 6602720 | Rec. |
| r 3 | 37 (37.0) | 0.86 | +1.57877 24450 25294 24451 28511 12243 022822E1 | Rec. |
| r 4 | 35 (35.8) | 0.81 | +6.76641 32385 94954 88492 78631 48650 6997E1 | Rec. |
| r 5 | 34 (34.7) | 0.79 | +3.31372 92591 18134 39512 67508 58932 271E2 | Rec. |
| r 6 | 33 (33.7) | 0.77 | +1.82640 38440 39425 39328 80354 49875 79E3 | Rec. |
| r 7 | 32 (32.7) | 0.74 | +1.11995 93049 36082 14972 18832 06923 0E4 | Rec. |
| r 8 | 31 (31.7) | 0.72 | +7.56777 88954 68002 96708 14587 25234E4 | Rec. |
| r 9 | 30 (30.8) | 0.70 | +5.58916 88019 33108 60490 38025 2233E5 | Rec. |
a) TE: total energy. VR: virial ratio.
b) NSF: number of significant figures. RNSF: real number of significant figures (written in parentheses).
c) Ratio of NSF to the NSF of TE.
d) "Rec." denotes the value recommended by the present study.
The He, Li, and Be atoms have common characteristics. Each ratio of the NSF of CC to the NSF of TE has almost the same value of 0.4. The NSFs of the radial expectation values are smaller than the NSF of TE. The ratio of the NSF of the radial expectation value, < ri > (i = −1, 1, ..., 9), to the NSF of TE becomes smaller for larger i. As seen from Figure 3, this ratio has an almost constant value independent of N.
We have performed multiple-precision HF calculations using 150 terms of the Λ function. This enables us to obtain TEs in high accuracy for the ground states of the He, Li, and Be atoms. Our recommended TE value for He (for example) is:
–2.8616799956 1223887877 5543740022 5634294128 7977675244 7053906864 69743
The NSFs of the TEs are 66, 43, and 43, for He, Li, and Be, respectively.
In our previous quadruple-precision calculations [32], the NSFs of CC were 16, 17, and 17 for He, Li, and Be respectively, but in the present multiple-precision calculation, they are 30, 19, and 19. The ratios of the NSF of CC to the NSF of TE are 0.45, 0.44, and 0.44, respectively, which are almost the same for all the three atoms. On the other hand, for Ar, Kr, Xe, Rn, and Og, our previous quadruple-precision calculations [32] showed that the NSFs were determined merely by the number of expansion terms, and would not be changed even if multiple-precision arithmetic was applied. For these five heavy atoms, the ratios of the NSFs of CC to those of TE are 0.47, 0.39, 0.45, 0.35 and 0.40 respectively. The present multiple-precision calculation indicates that the ratio of about 0.4 is generally valid for atoms from He to Og.
The authors thank Professor Emeritus Hiroshi Tatewaki of Nagoya City University for a critical reading of the manuscript. The main computation was performed using a cluster of Intel Xeon-based computers located at the Institute for Advanced Studies in Artificial Intelligence (IASAI) at Chukyo University.
