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Are Hartree–Fock atoms too small or too large?

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

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

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

As a measure, we use 〈r²〉.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

The unrestricted Hartree–Fock method is used.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

CASSCF calculations are also reported.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

Very large basis sets are employed.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj4

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj5

Introduction

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

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

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’.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

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

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot2

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac14

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac16

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac18

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

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot3

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

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj6

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj7

Results from these investigations are given in section 3.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac19

The tail behaviour of the wavefunction gives information on size.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac20

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

Indeed this analysis agrees with later results in this paper.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

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.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac21

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac22

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac23

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac24

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac25

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac26

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac27

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac28

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac29

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac30

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac32

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac33

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac34

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac35

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac36

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

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp2

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

Type: Method | Advantage: None | Novelty: New | ConceptID: Met1

We therefore perform investigations using the LDA and BLYP functionals.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj8

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj9

Hartree–Fock and density functional calculations

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod8

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

All orbitals are real.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

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).

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj10

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod13

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj11

CASSCF, CASPT2, and full CI calculations

Details of the calculations

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod16

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod16

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod17

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod19

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod20

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod22

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod23

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod24

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod26

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj12

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs7

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8

Analysis of the results

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj13

100

The appropriate atoms for this comparison were listed above.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac37

101

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

102

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

103

For the other atoms CASSCF is not identical to ROHF.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs10

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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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

105

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs11

106

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs12

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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs13

108

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs14

109

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs15

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

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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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

112

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

113

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj14

114

This is quite successful.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs16

115

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs17

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs18

117

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs19

118

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

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The effect is particularly pronounced for transition metal atoms.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

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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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs20

121

We conclude that dynamic correlation reduces atomic size.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

123

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

Summary

124

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac38

125

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

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).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

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〉).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

128

But these differences are small.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

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).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

130

Angular correlation therefore reduces atomic size.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con5

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

133

In the third row radial correlation only dominates for Kr.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

134

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con6

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con7

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).

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con8

137

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con9