1
Are Hartree–Fock atoms too small or too large?

2
We address the simple question, whether Hartree–Fock atoms are smaller or larger than exact (Schrödinger) atoms.

3
As a measure, we use 〈r2〉.

4
We study the ground state of the atoms He–Kr.

5
The unrestricted Hartree–Fock method is used.

6
To obtain the Schrödinger values we use the finite field CASPT2 approach, and the full CI scheme where possible.

7
CASSCF calculations are also reported.

8
Very large basis sets are employed.

9
We find that for those atoms for which the CASSCF wavefunction is distinct from the HF wavefunction, the Schrödinger values are distinctly smaller than the HF values.

10
Most other atoms are also smaller than the HF values, (or the same within a numerical uncertainty), the exceptions being the hard atoms N–Ne, and Cl, Ar and Kr.

11
We interpret our results in terms of categories of correlation contribution.

12
It is also of interest to test the performance of density functional theory (DFT); we find on the whole that the predictions are good, with B3LYP giving close agreement with finite field CASPT2 for nearly all atoms, in particular for the transition metals.

Introduction

13
One of the outstanding problems of computational quantum chemistry is the understanding of electron correlation.

14
The correlation energy is defined as the difference between the Hartree–Fock energy and the exact (non-relativistic) energy of the molecule.1

15
It is often separated into two parts, a short range effect due to the reduction in probability as two electrons come together, and a long range effect which is often responsible for electrons proceeding correctly to their atoms as the molecule dissociates.

16
The first part is called ‘dynamic correlation’, and is often associated with the necessity of including the interelectronic distance rij in accurate wavefunctions.

17
The importance of the electron–electron cusp is a dynamic correlation effect.

18
Quantum chemical calculations often introduce dynamic correlation through the introduction of ‘double replacement configurations’ into a configuration interaction (CI) expansion, although excluding double replacements into anti-bonding or non-bonding orbitals.

19
Dynamic correlation is often thought to be associated with a reduction in size because it allows electrons to arrange themselves in the most efficient manner possible.

20
The second part is called ‘left–right correlation’ for obvious reasons, but it is equally well called ‘static correlation’ or ‘near-degeneracy correlation’ or ‘non-dynamic correlation’.

21
Quantum chemists introduce this correlation through the introduction of double replacement configurations which involve anti-bonding or non-bonding or other valence orbitals.

22
Such correlations are responsible for an increase in molecular size: indeed for diatomic molecules, Hartree–Fock (HF) often predicts too short bonds, with CI calculations correctly lengthening them.

23
We deduce that this correlation has a greater effect on molecular shape than dynamic correlation.

24
The great difficulty is that in quantum chemistry these two forms of correlation cannot be cleanly separated.

25
For H2, the dominant double replacement CI configuration is σ2g → σ2u, which is clearly a left–right effect, but as the bond length decreases it becomes 1s2 → 2p2 in united atom notation, which is a dynamic effect.

26
It is often proclaimed that the correlation in atoms is all dynamic, because there are no long-range correlations.

27
The assignment is not clear for Be for which the near-degeneracy configuration 1s22p2 is very important, and this is commonly called ‘static correlation’.

28
Alternatively it may be argued that this configuration introduces r2ij and it is therefore a dynamic effect.

29
We recognise that angular correlation can be interpreted as an effect of r2ij.

30
Similar arguments can be presented for other atoms, where (ns)2→(np)2 are possible, for example the carbon atom ((2s)2 (2p)2,3P→(2p)4,3P), and the transition metals with a filled ns shell in the ground state.

31
Intuitively one might think these Hartree–Fock atoms are too large, because angular correlation should make them smaller (less screening).

32
On the other hand radial correlation could make them bigger, since one of the electrons moves to the next shell.

33
It is therefore relevant to ask the question whether HF atoms are too large or too small.

34
We attempt to ascertain the true size of atoms by performing CASSCF,2 CASPT23 and full CI (FCI) calculations using large basis sets.

35
We study the ground states of all atoms in the first three rows.

36
In all our investigations we shall calculate 〈∑i r2i〉, r2i = x2i + y2i + z2i, as a measure of atomic size.

37
Results from these investigations are given in section 3.

38
The tail behaviour of the wavefunction gives information on size.

39
The asymptotic behaviour of the exact density is exp(−2(2I)1/2r), whereas the asymptotic behaviour of the HF density is exp(−2(−2εhomo)1/2r).

40
If −εhomo>I, then the HF atom may be too small and vice-versa.

41
Indeed this analysis agrees with later results in this paper.

42
However we doubt whether the tail behaviour tells us about size because the emphasis should be on those regions where the electron density is high.

43
There are more accurate values of 〈r2〉 for selected small atoms, but they will not give a systematic comparison.

44
Density functional theory (DFT) introduces correlation effects rather differently.

45
The correlation functionals for the uniform electron gas, ascribed to Volko, Wilk and Nusair (VWN)4 or Perdew and Wang (PW91)5 are called dynamic correlation functionals, because there are no near-degeneracy configurations in that model system.

46
Perdew has extended his functional to include density gradients (P91).6

47
The functional LYP7 is also called a dynamic functional, because it was derived from the He correlated wavefunction of Colle and Salvetti (CS),8 basing one’s argument on the fact that all correlation in that atom must be dynamic.

48
It is also of the generalised gradient approximation (GGA) form.

49
It is well-known from many molecular calculations that these functionals do not introduce all molecular correlation; for example for N2, it is found9 that LYP (or P91) only introduces ∼50% of the necessary correlation for molecular binding.

50
Furthermore self-consistent Kohn–Sham (KS) calculations with HF+LYP (HFLYP) predict bondlengths which are even shorter than HF bondlengths.

51
The remarkable success of DFT hinges upon the fact that local exchange functionals introduce a localised exchange hole, which is close to the reference electron, in contradiction to the HF model which favours delocalised exchange holes.

52
The most popular local exchange functionals are the original uniform electron gas functional due to Dirac (LDAX) and the GGA exchange functional of Becke (B88X).10

53
The functional B88X introduces the other ∼50% of the correlation for the binding of N2; for these reasons it is said to introduce left–right correlation.

54
Such an analysis has been shown to work in practice for many hundreds of molecules.

55
Calculations with BLYP often successfully lengthen the too-short HF bondlengths.

56
The geometrical effects of the exchange functional are more significant than those of the dynamic functional.

57
B88X contains one parameter β, which was determined so that the exchange energies of the atoms He, Ne, Ar, Kr, Xe at the HF level were optimally reproduced.

58
For these noble gas atoms therefore, Becke believed that all the correlation should be dynamic.

59
Therefore one can ask the question, is dynamic or non-dynamic correlation more important for the determination of atomic size, in this DFT terminology?

60
Because the GGA functional BLYP is known to be good, we can analyse separately the contributions from B88X and LYP.

61
We therefore perform investigations using the LDA and BLYP functionals.

62
We first perform Hartree–Fock calculations and Kohn–Sham calculations, because of their similarity and simplicity.

Hartree–Fock and density functional calculations

63
For these single reference (one electron in each spin orbital) calculations, we have chosen to use the basis set which yields the lowest Hartree–Fock energy.

64
For all the calculations in this section we have used the unrestricted algorithm, because this is the proper procedure for KS calculations (see for example Pople et al11.), and therefore for comparison we should compute UHF energies.

65
The basis set used is due to Partridge12, which of course is appropriate for the occupied orbitals only.

66
All calculations were performed within CADPAC, with ∫ ρ(r)r2 dr being calculated using high radial quadrature.

67
The open shell atoms in many cases are not spherically symmetrical (eg C 2pαx 2pαy), but the values obtained would be the same if the atom was spherically symmetrised.

68
All orbitals are real.

69
We perform calculations on the atoms H-Kr; relevant ground states are Sc(d1s2,2D), Ti(d2s2,3F), V(d3s2,4F), Cr(d5s1,7S), Mn(d5s2,6S), Fe(d6s2,5D), Co(d7s2,4F), Ni(d8s2,3F), Cu(d10s1,2S), Zn(d10s2,1S).

70
In Fig. 1 we plot percentage ratio values of 〈∑ir2i〉 (denoted as r2 in our discussions) for the local exchange LDAX (Dirac) and the GGA exchange B88X10 functionals, compared to the UHF values.

71
In Fig. 2 we show the percentage ratio for the local LDA (LDAX+VWN) and the GGA BLYP exchange–correlation functionals.

72
Fig. 1 shows that the LDAX atoms are larger than the B88X atoms, for H through Ar, but for the heavier atoms there is little percentage difference.

73
B88X atoms are larger than UHF atoms for H-Li, B-Ne, Al–Ar, Cr, Ga–Kr.

74
The introduction of dynamic correlation through the correlation functionals reduces atomic size, see Fig. 2.

75
We see that LDA and BLYP atoms are similar in size; only He, C–Ne, BLYP atoms are substantially larger than UHF atoms, with S-Ar and Se-Kr marginally larger.

76
DFT is suggesting that most Schrödinger atoms are smaller than UHF atoms, with the clear exception of the hard first row atoms, and He, P–Ar, Se–Kr.

77
It will be seen below that these predictions are in agreement with those from ab initio quantum chemistry calculations.

78
These we now report, and then in the final section we will compare the two sets of calculations, and give our explanation for the results.

CASSCF, CASPT2, and full CI calculations

Details of the calculations

79
Calculations have been performed using ANO-L basis sets13 for the atoms He-Zn with the exception of K and Ca for which a relativistic basis set (ANO-RCC)14 was used (designed for use with the Douglas–Kroll Hamiltonian).

80
Slightly smaller values of 〈r2〉 were consequently obtained for these two atoms.

81
The ANO-L basis sets were not available for Ga–Kr.

82
Instead, a smaller ANO-S basis set15 was used.

83
Comparison of the HF results shows that also this basis set is adequate for the present purpose.

84
The details of the basis sets are: AtomsPrimitiveContractedHe9s4p3d2f5s4p2dLi–Ne14s9p4d3f6s5p3d1fNa–Ar17s12p5d4f7s6p4d2fK21s16p5d4f10s9p5d3fCa20s16p6d2f10s9p6d2fSc–Zn21s15p10d6f4g8s7p5d3f2gGa–Kr17s15p9d9s9p5d

85
The active space used in the CASSCF calculations2 comprises all valence orbitals: 2s,2p for atoms Li–Ne, 3s,3p for Na-Ar, 4s,4p for K–Ca, 3d,4s,4p for Sc–Zn and 4s,4p for Ga–Kr.

86
CASSCF is therefore identical to restricted open-shell HF (ROHF) for the atoms: He, Li, N–Ne, Na, P–Ar, K, Cr, Cu, and As–Kr.

87
Spherical symmetry is assured by performing the calculations in Ci symmetry and averaging over the components of a given term.

88
The wave functions are of course eigenfunctions of the spin operator Ŝ2.

89
The CASPT2 calculations3 are performed using the G1 correction to the zeroth order Hamiltonian.

90
A small imaginary level shift (0.05) was used for the transition metals.

91
The outermost core electrons were included in the correlation treatment (specifically 1s for Li–B, 2s,2p for Na-Ar, 3s–3p for K–Zn, and 3d for Ga–Kr).

92
The 〈r2〉 values have been obtained using finite field perturbation theory (FFPT).

93
The method was calibrated by controlling it to give the same result as the expectation value for CASSCF wavefunctions.

94
Full CI (FCI) calculations were also performed for the atoms He–N and Mg–P.

95
The results are quite similar to FFPT/CASPT2, which is reassuring.

96
The calculations have been performed using the MOLCAS-6.0 software.

97
The percentage ratios for FFPT/CASPT2 (denoted PT2) r2 values compared to UHF have been given in Fig. 2.

98
Fig. 3 gives the percentage ratios for FFPT/CASPT2 and the hybrid functional B3LYP16.

Analysis of the results

99
In order to calibrate the present results with those obtained using UHF and the Partridge basis set in section 2, we first compared the r2 values with the ROHF results obtained using the CASSCF method.

100
The appropriate atoms for this comparison were listed above.

101
There were no notable differences for the atoms Li–Ar and As–Kr, (indeed less than 0.05 a20).

102
The difference is marginally larger for K (0.20 a20), which could be due to the different basis sets used.

103
For the other atoms CASSCF is not identical to ROHF.

104
The reason is that in these cases there are important intra-valence correlation effects due to excitations from the doubly occupied ns shell to the empty (for the transition metals) or partially empty (for the main group atoms) np shell.

105
We have collected the results for these atoms in the Table 1.

106
Also the the hybrid DFT functional B3LYP16 and the FFPT/CASPT2 results are presented in this table.

107
(For completeness we give the FFPT values for the atoms not in the table: H 3.00, He 2.39, Li 18.45, N 12.20, O 11.39, F 10.49, Ne 9.62, Na 26.65, P 30.09, S 29.12, Cl 27.68, Ar 26.13, K 47.64, Cr 36.60, Cu 31.01, As 40.74, Se 41.04, Br 40.48, Kr 39.55).

108
We now see a clear difference between the UHF and CASSCF results.

109
The latter are constantly smaller and the difference is larger the more empty the p-shell.

110
This difference was anticipated in the introduction and can be easily understood.

111
The s2 → p2 excitation introduces angular correlation between the s-electrons, which increases the probability to find the two electrons in different sides of the atom.

112
The separation of the electron pair leads to a decreased screening of the core and the atom shrinks.

113
The DFT functionals attempt to correct for this error in the UHF density.

114
This is quite successful.

115
The atom-sizes for BLYP are close (in the majority of cases) to the results obtained with FFPT/CASPT2.

116
Comparing to the full CI results we see that DFT/B3LYP slightly overcompensates (except for Mg), while the FFPT/CASPT2 results are quite similar to the FCI results.

117
Indeed the B3LYP predictions are very close (in magnitude) to the FFPT/CASPT2 predictions.

118
Finally, the overall effect of static correlation is to decrease the atomic size!

119
The effect is particularly pronounced for transition metal atoms.

120
However both atoms with large intra-valence correlation effects (eg V) and those with none (eg Cr) decrease in size on including dynamic correlation effects through introducing PT2 to CASSCF.

121
We conclude that dynamic correlation reduces atomic size.

122
Therefore both the static and the dynamical correlation effects lead to a decreased size of the atoms.

123
The effect is largest for atoms with large static correlation effects.

Summary

124
Within the scope of the present calculations, we are now able to answer the posed questions.

125
Firstly, most Schrödinger atoms are smaller than the UHF atoms.

126
For He, Li, C, S, Cl, Ar, Se, Br and Kr the sizes (〈r2〉) are the same within numerical accuracy (estimated to be 0.1 a20).

127
N (0.13), O (0.17), F (0.24) and Ne (0.25) are indicated to have larger Schrödinger atoms than UHF atoms (the values in parenthesis are the differences in 〈r2〉).

128
But these differences are small.

129
In the cases for which the CASSCF wavefunction is distinct from the HF wavefunction, the Schrödinger atoms are significantly smaller than the HF atoms (see the table, C is the exception).

130
Angular correlation therefore reduces atomic size.

131
For the hard atoms N–Ne, there is no s2 → p2 excitation, and thus radial correlation is more important, leading to an increase in size for these atoms.

132
For the second row it is probable that angular correlation is introduced through the 3d orbitals, leading to the difference with the first row atoms, but radial correlation becomes dominant for Cl and Ar.

133
In the third row radial correlation only dominates for Kr.

134
The summary is that these size differences reflect the balance between angular and radial correlation effects.

135
The predictions from DFT are in general in good agreement with the predictions from FFPT/CASPT2.

136
In particular the predictions with B3LYP are within 0.3 a20 for nearly all atoms (exceptions are Li, Be, B, Na, Mg, Ca and Cr), in particular the transition metal atoms are very well predicted (except Cr).

137
The predictions with BLYP are a little inferior, but they are superior to the predictions of CASSCF, as might be expected.