J. Phys. B: At. Mol. Opt. Phys. 32 (1999) 2245–2255. Printed in the UK
PII: S0953-4075(99)00845-7
Correlated one-body momentum density for helium to neon
atoms
A Sarsa, F J Gálvez and E Buendı́a
Departamento de Fı́sica Moderna, Facultad de Ciencias, Universidad de Granada, E-18071
Granada, Spain
Received 12 January 1999
Abstract. Starting from explicitly correlated wavefunctions, the one-body momentum density,
γ (p),
E and the expectation values hδ(p)i
E and hpn i, with n = −2 to +3, have been obtained for the
atoms helium to neon. All the calculations have been carried out by using the Monte Carlo algorithm.
An analysis of the numerical accuracy of the method has been performed within the Hartree–
Fock framework. The effects of the electronic correlations have been systematically studied by
comparing the correlated results with the corresponding Hartree–Fock ones.
1. Introduction
Position and momentum space properties provide complementary information to describe the
structure of electronic systems. An important quantity in momentum space is the atomic
one-body electron density, γ (p),
E defined as
γ (p)
E = h8(pE1 α1 , . . . , pEN αN )|
N
X
δ[pE − pEj ]|8(pE1 α1 , . . . , pEN αN )i
(1)
j =1
where αi is the spin coordinate and 8(pE1 α1 , . . . , pEN αN ) stands for the momentum space
wavefunction. This quantity is the probability density function for an electron having a
momentum p.
E Some of its radial expectation values, hp n i, have a special physical significance.
The value hp −1 i is related to the height of the peak of the Compton profile, hp2 i gives us twice
the negative of the total energy of the atom due to the virial theorem, and hp4 i is related to
relativistic corrections. In addition, one can obtain from γ (p)
E the atomic Compton profile,
within the impulse approximation, which is an experimentally measurable quantity [1]. We
shall denote its spherical average by 5(p).
Different calculations of the one-body momentum density have been carried out within
the Hartree–Fock framework [2–4] leading to extensive tabulations of Hartree–Fock atomic
momentum-space properties [5–9]. However correlated momentum space densities are more
difficult to calculate. For two electron systems, γ (p)
E has been obtained from explicitly
correlated wavefunctions [10–12]. For atoms with three or more electrons, explicitly correlated
wavefunctions have not been used to obtain γ (p)
E due to the difficulties which appear in solving
the different integrals involved in its evaluation. For them the calculation has been performed
in a configuration interaction (CI) scheme for the lithium [13], beryllium [14] and neon [15]
isoelectronic series. Recently [16], γ (p)
E has been obtained for the atoms helium to argon
working in this same scheme but expanding the single-particle wavefunctions in a basis set of
Gaussian-type orbitals.
0953-4075/99/092245+11$19.50
© 1999 IOP Publishing Ltd
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A Sarsa et al
As is known, the Monte Carlo method allows one to evaluate the expectation value of any
operator between wavefunctions of any type, becoming a powerful tool in quantum chemistry
calculations [17]. Thus one can calculate different atomic properties by using an explicitly
correlated wavefunction instead of using a large expansion in Slater determinants. The main
aim of this work is, by using this method and working with explicitly correlated wavefunctions,
to obtain local and global properties of the single-particle momentum density for the atoms
helium to neon. Then an analysis of the effects of the electronic correlations on these quantities
is carried out by comparing the correlated values with the corresponding Hartree–Fock ones.
The structure of this paper is as follows. In section 2 we show the expressions used to obtain
the γ (p)
E density and the method used to calculate its moments hpn i. In section 3 we describe
the functional form of the wavefunctions we have worked with, giving their main properties. In
section 4 we present the results obtained for the correlated single-particle momentum density,
and, finally, in section 5 we give the main conclusions. Atomic units are used throughout this
paper.
2. Monte Carlo method and momentum distributions
The one-body momentum density may be built from the momentum wavefunction but this
is, in general, unknown if the position wavefunction depends explicitly on the interelectronic
coordinates. Thus, to use the Monte Carlo method in these cases, we have expressed the
momentum density in terms of the position wavefunction as
(
)
Z
ip·(E
E r1 −Er10 )
Nα X
e
dτr dEr10 1 (τr , rE10 α10 )
E =
δαα1 δαα10
γα (p)
(2)
3
h9|9i
(2π ) 2
E is the contribution to γ (p)
E of the electrons with spin α, τr accounts for the spatial,
where γα (p)
rE1 , . . . , rEN , and the spin, α1 , . . . , αN , coordinates, the integral symbol means integration over
rE1 , . . . , rEN , and rE10 , and the summation symbol means sums over α1 , . . . , αN , and α10 , Nα is the
number of electrons with spin α, and
1 (τr , rE10 α10 ) = 9 ∗ (Er1 α1 , rE2 α2 , . . . , rEN αN )9(Er10 α10 , rE2 α2 , . . . , rEN αN ).
(3)
E by means of the Monte Carlo method it is useful to use the identity
To calculate γα (p)
Z
Z
1 X
1
0
0 −r10
dEr1 e h9|9i =
dτr dEr10 e−r1 |9(τr )|2 .
(4)
h9|9i =
8π
8π
0
The function in the integrand can be chosen in different ways, but this particular choice (e−r1 )
is very appropriate in studying the low-p region in several atoms [18]. The use of equation (4)
allows us to write equation (2) as
R
dτ ω(τ )f (τ )
hf ; ωi
hf : g; ωi = RV
(5)
=
hg; ωi
V dτ ω(τ )g(τ )
where ω(τ ) stands for the distribution function used in the Monte Carlo sampling. We have used
two different distribution functions in this paper. The first one has been chosen as |1 (τr , rE10 α10 )|
and the results obtained with it have been labelled as MC1 . This distribution function works
adequately except for describing the low-p region in the atoms beryllium, boron and carbon.
The second one has been taken as exp[−r10 ]|9|2 and has been introduced to improve the results
in the low p region for beryllium, boron and carbon atoms. The results obtained when this
distribution function is used are labelled as MC2 . A more detailed description is given at the
end of this section.
In general the Monte Carlo algorithm is able to calculate 5(p) (with a relative error less
than or equal to 0.2 or 0.3%) up to a value pmax 0 where it shows an oscillating behaviour
Correlated one-body momentum density for helium to neon atoms
2247
around the exact density [19] (for example, for neon within the Hartree–Fock framework
and working with 108 movements for each one of the electrons the oscillations appear for
p ' 14 au). This behaviour means that the moments hpn i defined as
Z
hp n i =
Z
dpE pn γ (p)
E =
dpE pn 5(p)
(6)
cannot be accurately obtained from the Monte Carlo calculation of 5(p). To avoid this problem
we have numerically integrated until the point pmax where the density is well determined and
then, from this point to ∞, the integral has been carried out by using the asymptotic behaviour
of the momentum distribution given by [20]
5(p) =
C8 C10 C12
+
+
+ ···.
p8 p10 p12
(7)
To approximate the asymptotic behaviour of 5(p) we have used the coefficients C8 , . . . , C14 .
The coefficients Cj are fixed from the data in an interval [pmin , pmax ] by fitting the asymptotic
expression (7) to the corresponding Monte Carlo density. To select the best interval, i.e. the best
value of pmin , we have minimized the difference between the calculated and the exact values
of both the norm and the expectation value hp2 i. The former is the number of particles and
P 2
E i i in position space. With this method
the latter is obtained as the expectation value −h i 5
the exact norm of the momentum density is not obtained. For example, we have obtained
hp 0 i = 2.999 91 and 4.998 for lithium and boron, respectively, which gives us information
about the accuracy in the results we can expect. Then, once we have obtained these results, and
in order to calculate all the rest of the moments, we have normalized the momentum density
to the number of electrons. Let us finally comment that the coefficients Cj obtained from the
different correlated wavefunctions have alternate sign, as happened within the Hartree–Fock
framework [5] and by using CI wavefunctions [15].
The method used here to determine the different momentum properties has been tested
within the Roothaan–Hartree–Fock framework by using the wavefunctions tabulated by
Clementi and Roetti [21]. In table 1 we show, for the atoms helium to boron, the values
of 5(0) and the expectation values hp n i, n = −2, −1, 1, 2 and 3, obtained with the Monte
Carlo quadrature (MC1 row) as compared with the analytic results of Garcı́a de la Vega and
Miguel [6]. For helium and lithium the values of the density at the origin and the moments
of negative order are quite well reproduced with the Monte Carlo method. However, for
the beryllium atom the one-body momentum density is underestimated for low values of the
momentum p as can be noticed from the values of 5(0) and the moments of negative order.
This bad behaviour also holds for boron and carbon. To improve the results in that region
we have used the second distribution function mentioned above. This distribution function
amends not only the values of 5(p) for low values of the argument but also the behaviour
of the intracule density in momentum space for low values of the interelectronic momentum,
p12 [22]. However the momentum density obtained with it starts to oscillate for smaller pvalues than in the other case. It is possible to exploit the correct behaviour of each one of
the momentum densities obtained with the use of the corresponding distribution function by
connecting them at an intermediate point. The results so obtained for beryllium and boron are
also shown in table 2 (MC2 row). The improvement in 5(0) and in the different momentum
expectation values, mainly in hp−2 i and hp−1 i, is apparent. The atoms nitrogen to neon show
the same behaviour as helium and lithium.
2248
A Sarsa et al
Table 1. Comparison between the analytic and the Monte Carlo values of several one-body
momentum properties within the Hartree–Fock framework for the atoms helium, lithium, beryllium
and boron. The rows MC1 and MC2 stand for two different distribution functions used in the Monte
Carlo sampling. See text for more details.
5(0)
hp−2 i
hp−1 i
hpi
hp 2 i
hp 3 i
He (1 S)
MC1
[6]
0.438 4
0.439 0
4.084
4.0893
2.139
2.140 58
2.798
2.799 02
Li (2 S)
MC1
[6]
8.57(3)
8.567 3
26.54
26.5566
5.181
5.185 9
4.92
4.905 6
14.865 6
14.865 5
Be (1 S)
MC1
MC2
[6]
5.86(2)
5.943(7)
5.950 77
24.97
25.28
25.290
6.260
6.317
6.318 24
7.49
7.43
7.434 20
29.143
29.147
29.146 1
179.0
187.7
185.588
B (2 P)
MC1
MC2
[6]
2.48(4)
2.521(6)
2.535 1
16.17
16.25
16.2617
5.958
5.983
5.979 52
49.100
49.030
49.057 9
375
395
383.697
10.70
10.61
10.649 4
5.724
5.723 43
17.84
17.9909
69.30
70.9948
3. Correlated wavefunction
The structure of the correlated wavefunction used in this work, 9, is the product of a symmetric
correlation factor, F , which includes the dynamic correlation among the electrons times a
model wavefunction, 8, that provides the correct properties of the exact wavefunction such
as the spin and the angular momentum of the atom, and is antisymmetric in the electronic
coordinates:
9 = F 8.
(8)
For the correlation factor we use the form of Boys and Handy [23]
F =e
P
i>j
Uij
(9)
with
Uij =
Nc
X
ck (r̄imk r̄jnk + r̄ink r̄jmk )r̄ijok
(10)
k=1
and
r̄i =
ri
,
1 + ri
r̄ij =
rij
.
1 + rij
(11)
The set of values for mk , nk and ok determines the parametrization selected for the generalized
Jastrow factor. In this paper we take the values proposed by Schmidt and Moskowitz [24] by
using arguments based on the requirement of local current conservation. The correlation factor
will have 7, 9 or 17 variational parameters corresponding to the configurations (mk , nk , ok ) =
(0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (2, 0, 0), (3, 0, 0), (4, 0, 0); (2, 2, 0), (2, 0, 2); (2, 2, 2),
(4, 0, 2), (2, 0, 4), (4, 2, 2), (6, 0, 2), (4, 0, 4), (2, 2, 4) and (2, 0, 6). The first seven include
electron–electron and electron–nucleus correlations, while the rest of the configurations take
into account the electron–electron–nucleus correlations.
Correlated one-body momentum density for helium to neon atoms
2249
Ground state energy and values of 5(0) and hp n i for the atoms helium, lithium, beryllium
Table 2.
and neon obtained with the different correlated wavefunctions used in this work as compared with
the Hartree–Fock results and with previous correlated ones. In parentheses we give the statistical
errors in the last figure obtained with the Monte Carlo sampling.
E
5(0)
He (1 S)
HF [9]
97
99
917
HY [12]
−2.861 68
−2.899 8(1)
−2.903 01(5)
−2.903 63(2)
−2.903 321
0.439 85
0.428 0
0.436 0
0.440 5
0.443 40
Li (2 S)
HF [9]
97
98
910
918
CI [13]
−7.432 7
−7.473 8(2)
−7.474 2(1)
−7.476 1(6)
−7.477 5(4)
−7.477 699
8.555 4
8.259 9(8)
8.182 5(6)
8.997 0(7)
7.058(7)
8.346 12
Be (1 S)
HF [9]
97
917,1
910,1
CI [14]
−14.573 0
−14.624 9(1)
−14.647 7(2)
−14.652 3(1)
−14.661 02
5.952 6
5.82(2)
6.60(2)
5.21(2)
4.840 64
Ne (1 S)
HF [9]
97
99
917
CI [15]
−128.547 1
−128.769(1)
−128.877(1)
−128.882(2)
−128.904 5
0.244 93
0.246 4(4)
0.256 7(5)
0.250 1(6)
0.250 73
hp −2 i
4.092 3
4.008
4.061
4.074
4.101 38
hp −1 i
hpi
hp 2 i
2.141 0
2.119
2.131
2.132
2.139 13
2.799 0
2.823
2.816
2.820
2.814 11
26.544
25.94
25.20
27.11
24.20
26.134
5.185 5
5.132
5.035
5.197
5.026
5.151 9
4.905 6
4.919
4.971
4.927
4.961
4.919 6
14.865
14.954
14.989
14.907
14.945
14.955
25.294
24.96
26.75
22.82
21.938 7
6.318 5
6.29
6.424
5.99
5.909 1
7.434 2
7.448
7.392
7.537
7.532 98
29.146
29.15
29.17
29.400
29.332 9
5.469 4
5.434
5.54
5.52
5.552 7
5.453 7
5.429
5.462
5.432
5.478 2
35.197
35.56
35.42
35.68
35.241 2
5.723 4
5.778
5.778
5.796
5.805 43
257.09
255.31
258.72
257.81
257.751
hp 3 i
17.990
17.82
17.99
18.09
18.3961
70.977
71.86
69.02
70.20
68.59
71.643
185.55
182.0
190.4
184.6
186.351
3584.2
3250.0
3508.0
3350.0
3591.5
The model wavefunction, 8, has been chosen in a variety of ways. In the first we have
taken the Roothaan–Hartree–Fock wavefunctions tabulated by Clementi and Roetti [21] and
the total wavefunction has been denoted by 97 , 99 and 917 . For beryllium, boron and carbon
we have also considered the 2s–2p near degeneracy effect [25] by using the multideterminant
wavefunction
8 = 81 + λ82
(12)
2
2
where 81 and 82 are the Hartree–Fock solutions corresponding to the configurations 1s 2s 2pk
and 1s2 2pk+2 , respectively, and k = 0, 1, 2 for beryllium, boron and carbon, respectively. Here
λ is a new variational parameter and the total wavefunction has been denoted by 9n,1 . Finally,
for lithium and beryllium, we have also considered the variation of 8 in the minimization
process. In doing so 8 does not satisfy the electron–nucleus cusp any more and therefore,
to retrieve this property of the total wavefunction, we have also included the configuration
(1, 0, 0) in the correlation factor, working with 8, 10 and 18 variational parameters. Thus, in
these cases, the total wavefunction has been denoted as 9n and 9n,1 (with n = 8, 10 and 18)
for lithium and beryllium, respectively.
The ground state energy and different properties provided by these wavefunctions can be
found in [19, 24, 26–28]. The best energy for each atom is obtained with 917 for helium and
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A Sarsa et al
for nitrogen to neon (although for the latter the wavefunction 99 provides practically the same
energy as 917 ), with 917 and 918 for lithium, with 917,1 and 910,1 for beryllium and, finally,
with 917,1 for boron and carbon. However a detailed study of both the one- and the two-body
densities in position space leads to the conclusions that 917 for lithium and 917,1 for beryllium
provide densities that are very extended in space and that, although 97 recovers about 90%
of the correlation energy for the lithium atom, it describes adequately both the single-particle
and the electron pair densities [19,27]. Besides, 99 and 917 , for the neon atom provide a quite
similar single particle density, even a bit better for the former for large values of the electron
coordinate.
4. Correlated single-particle momentum density
By using correlated wavefunctions we shall study both the spherically averaged momentum
density, 5(p), and the radial momentum density I (p) = 4πp 2 5(p). To analyse the effects
of the electronic correlation we shall study the difference function
1In = 4πp2 [5n (p) − 5HF (p)] = In (p) − IHF (p)
where 5n is the correlated momentum density obtained with any of the correlated
wavefunctions we are working with and 5HF is the Hartree–Fock density. We shall also
calculate the momentum density at the origin, 5(0), and the expectation values hpn i.
First we study the atoms helium, lithium, beryllium and neon, comparing our results with
others that we consider as exact. For the helium atom, the 1In functions obtained with 97 ,
99 and 917 are compared with the one obtained from a Hylleraas-type wavefunction [12]. As
we can see in figure 1, the functions 97 and 99 are not able to reproduce the first maximum,
and only 917 provides a quite small one. In spite of that, the functions seem to converge to
the exact one. In addition, the different expectation values reported in table 2 converge to the
exact ones. In particular, 917 reproduces adequately the results considered as exact.
For the lithium atom, the different correlated wavefunctions used in this paper do not
provide a set of convergent results in 5(p) for low values of the momentum, as can be noticed
from the values in table 2. In figure 1 we plot the difference function 1In for several correlated
wavefunctions. Despite 918 recovering around 99% of the correlation energy, it reproduces
only qualitatively the behaviour of the difference function 1I (p) as compared with the one
of [13] where a CI wavefunction with the same correlation energy was used. We have found
that only 97 (which provides about 90% of the correlation energy) is able to give acceptable
approximations of both the different expectation values shown in table 2 and the 1I (p) function
that can be considered as exact. At this point let us remark that both the one- and two-body
densities in position space calculated with this wavefunction are of quite good quality [19,27],
although 918 was considered as a better wavefunction because it provided both a much better
energy and also a quite good description of the electronic distribution. Thus the electron–
electron–nucleus correlations, which are present in 9n , n = 9, 10, 17 and 18, but not in 97 ,
distort the 5(p) distribution for small values of the momentum p in the lithium atom, not
being able to provide a convergent set of results for this density.
The system for which the effects of the electronic correlations are more important is
the beryllium atom. As we can see in figure 1, the wavefunction 910,1 , which takes into
account the electron–electron–nucleus dynamic correlation, the nondynamic one due to the
2s–2p near degeneracy, and has been determined by modifying the central part of the total
wavefunction, is the only one that reproduces with great precision the details of the difference
function considered as exact [14]. In addition, the different expectation values obtained with
this wavefunction and shown in table 2 approach those considered as exact although a plot of
Correlated one-body momentum density for helium to neon atoms
2251
Figure 1. Difference function, 1In , for the atoms helium, lithium, beryllium and neon obtained
from (a) 97 , 99 and 917 for helium, (b) 97 and 918 for lithium, (c) 97 , 917,1 and 910,1 for
beryllium and (d) 99 and 917 for neon. The results are compared with others, labelled as exact,
obtained from a Hylleraas-type wavefunction for helium, and from a CI calculation for lithium,
beryllium and neon.
the momentum density, 5(p), still does not superimpose on the one considered as exact for
low values of the momentum p. In spite of its simplicity, 910,1 is able to reproduce the most
important physical effects due to electronic correlations.
For the neon atom, we have calculated the momentum distribution with the wavefunctions
99 and 917 . The difference function derived from 99 is quite similar to that considered as
exact [15], as one can check in figure 1. At this point, let us remember that, for neon, 99 and
917 provide practically the same ground state energy [24] although the long-range behaviour
of the single-particle position density, that can be extracted from the expectation values hr n i,
n > 0, is a bit better determined by 99 [19]. Thus one can expect that the same happens
for the short-range behaviour in momentum space, because the Fourier transform relates the
wavefunctions in both spaces.
The effects of the electronic correlation in the one-body momentum density are well
described by using the wavefunction 917,1 , which takes into account the 2s–2p near degeneracy
[28], for boron and carbon, and 917 for nitrogen to fluorine. The values of 5(0) and hpn i
are shown in table 3, where they are compared with the Hartree–Fock ones and with those
obtained in a configuration interaction scheme [16]. In figure 2 we plot the difference function
1I (p) for these atoms obtained with the wavefunctions mentioned above. As one can notice,
the short-range p-behaviour of the difference function for boron and carbon is opposite to the
one found for nitrogen to fluorine. Thus the correlations decrease the momentum distribution
2252
A Sarsa et al
Table 3. As table 2, for the atoms boron to fluorine.
5(0)
hp−2 i
hp−1 i
hpi
B (2 P)
HF [9] −24.529 1
917,1 −24.632 4(2)
CI [16] −24.651 81
2.528 9
2.46(1)
16.252
15.87
15.0999
5.9791
5.899
5.7765
10.649
10.73
10.7496
49.058
49.277
49.3000
383.65
377.7
C (3 P)
HF [9] −37.688 6
917,1 −37.811 3(3)
CI [16] −37.841 57
1.336 1
1.328(1)
11.752
11.67
11.4272
5.7542
5.713
5.6699
14.462
14.628
14.5505
75.377
75.70
75.6790
691.44
660.0
N (4 S)
HF [9] −54.400 9
−54.545 6(3)
917
CI [16] −54.582 43
0.797 26
0.813(2)
9.0970
9.06
9.1267
5.5973
5.534
5.5839
18.863
19.37
18.9313
108.80
109.0
109.1604
1136.2
990.0
O (3 P)
HF [9] −74.809 4
−75.014 6(1)
917
CI [16] −75.056 99
0.506 60
0.514(2)
7.4893
7.51
7.5523
5.5507
5.511
5.5575
23.721
24.13
23.7865
149.62
149.55
150.1015
1746.3
1600.0
F (2 P)
HF [9] −99.409 3
−99.674(1)
917
CI [16] −99.719 33
0.343 90
0.367 0(7)
6.3365
6.63
6.3981
5.5016
5.566
5.5153
29.166
29.41
29.2297
198.82
199.77
199.4362
2551.8
2500.0
E
hp2 i
hp3 i
Figure 2. As figure 1, for the atoms boron to fluorine.
Correlated one-body momentum density for helium to neon atoms
2253
Table 4. Expectation values of some differential operators for the atoms helium to neon obtained
from the Hartree–Fock solution and from a correlated wavefunction. In parentheses we give the
statistical error in the last digit. For oxygen to neon we also show some characteristics of the
momentum densities.
P 2
P 4
Ei i
Ei i
−h i 5
h i5
p0
5(p0 )
He (1 S)
HF
917
5.7234
5.796(5)
105.63
106(1)
Li (2 S)
HF
97
14.865
14.97(2)
622.60
620(10)
Be (1 S)
HF
910,1
29.146
29.40(3)
2 158.7
2 140(30)
B (2 P)
HF
917,1
49.058
49.24(3)
5 537.8
5 320(70)
C (3 P)
HF
917,1
75.377
75.7(1)
11 870
11 800(200)
N (4 S)
HF
917
108.80
109.5(2)
22 533
22 100(400)
O (3 P)
HF
917
149.62
149.6(2)
39 207
38 000(1000)
0.3001
0.31
0.5200
0.539
F (2 P)
HF
917
198.82
199.8(3)
63 814
63 000(1000)
0.4443
0.41
0.3771
0.429
Ne (1 S)
HF
99
257.09
258.9(4)
98 580
102 000(3000)
0.5528
0.53
0.2871
0.297
for low values of p in the former and increase it for the latter.
P 2
P 4
E i i and h i 5
E i i for the best
In table 4, we show the correlated expectation values h i 5
correlated wavefunctions used in this paper as compared with the Hartree–Fock ones [9]. The
correlated expectation value hp 4 i compares quite well with previous calculations for all the
atoms considered. Our results differ from those obtained by Alexander and Coldwell [26] with
917 for those systems which contain p-type orbitals.
A general result for all the atoms here studied is that the electronic correlations are
important in the description of the one-body momentum distribution but they do not modify
its properties of monotonicity found within the Hartree–Fock framework [2–4]. In particular,
atoms helium to nitrogen have a monotonically decreasing momentum density, while atoms
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A Sarsa et al
oxygen to neon have a momentum density with a local minimum at p = 0 and a maximum at
p0 > 0. For these three atoms we have compared in table 4 the correlated values of p0 and
5(p0 ) with the Hartree–Fock ones of [4]. As can be seen, the effect of electronic correlations
is to shift the position of the maximum and to increase its value. This effect is more important
in the fluorine atom.
5. Conclusions
By using the Monte Carlo algorithm, we have calculated the one-body momentum density
and its radial moments for the atoms helium to neon starting from the explicitly correlated
wavefunctions of Schmidt and Moskowitz [24] and a generalization of them to include the
nondynamic effect due to the 2s–2p near degeneracy in the atoms beryllium, boron and carbon.
We have also modified the central part of the wavefunction in the minimization process for the
lithium and beryllium atoms.
The Monte Carlo method allows the numerical problems involved in the determination of
the single-particle momentum properties to be solved. In those atoms for which precise data
are available, the wavefunctions used in this work reproduce the effects obtained with more
sophisticated wavefunctions.
For the helium atom the wavefunctions 97 , 99 and 917 provide a set of convergent results
that lead to a good description of the one-body density. The lithium atom is a very difficult
system to reproduce. The best results are obtained with 97 , which only includes electron–
electron and electron–nucleus correlations. This type of wavefunction has been found to work
adequately for light atoms [29]. The beryllium atom is the one for which the correlations
show more important effects on the one-body momentum density; the wavefunction used to
describe it, which includes electron–electron–nucleus correlations and takes into account the
2s–2p near degeneracy (the mean-field wavefunction has also been obtained in the optimization
process), allows one to get a 1I (p) function that fits the one that can be considered as exact.
For the other atoms, the results reproduce qualitatively the ones obtained in previous works.
The wavefunctions used allow previous calculations to be reproduced adequately, including
the expectation value hp 4 i for atoms that contain p-type orbitals.
Acknowledgments
This work has been partially supported by the Spanish Dirección General de Investigación
Cientı́fica y Técnica (DGICYT) under contract PB95-1211-A and by the Junta de Andalucı́a.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Williams B G 1977 Compton Scattering (New York: McGraw-Hill)
Simas A M, Westgate V M and Smith V H Jr 1984 J. Chem. Phys. 80 2636
Westgate V M, Simas A M and Smith V H Jr 1985 J. Chem. Phys. 83 4054
Koga T, Matsuyama H, Inomata H, Romera E, Dehesa J S and Thakkar A J 1998 J. Chem. Phys. 109 1601
Thakkar A J, Wonfor A L and Pedersen W A 1987 J. Chem. Phys. 87 1212
Garcı́a de la Vega J M and Miguel B 1993 At. Data Nucl. Data Tables 54 1
Garcı́a de la Vega J M and Miguel B 1994 At. Data Nucl. Data Tables 58 307
Garcı́a de la Vega J M and Miguel B 1995 At. Data Nucl. Data Tables 60 321
Koga T and Thakkar A J 1996 J. Phys. B: At. Mol. Opt. Phys. 29 2973
Benesch R 1973 J. Phys. B: At. Mol. Phys. 9 2587
Regier P E and Thakkar A J 1985 J. Phys. B: At. Mol. Phys. 18 3061
Arias de Saavedra F, Buendı́a E and Gálvez F J 1996 Z. Phys. D 38 25
Correlated one-body momentum density for helium to neon atoms
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
2255
Esquivel R O, Tripahi A N, Sagar R P and Smith V H Jr 1991 J. Phys. B: At. Mol. Opt. Phys. 25 2925
Tripahi A N, Sagar R P, Esquivel R O and Smith V H Jr 1992 Phys. Rev. A 45 4385
Tripahi A N, Smith V H Jr, Sagar R P and Esquivel R O 1996 Phys. Rev. A 54 1877
Meyer H, Müller T and Schweig A 1996 J. Mol. Struct. (Theochem.) 360 55
Hammond B L, Lester W A Jr and Reynolds P J 1994 Monte Carlo Methods in Ab Initio Quantum Chemistry
(Singapore: World Scientific)
Sarsa A, Gálvez F J and Buendı́a E 1999 Comput. Phys. Commun. at press
Sarsa A 1998 PhD Thesis University of Granada
Kimball J C 1975 J. Phys. A: Math. Gen. 8 1513
Clementi E and Roetti C 1974 At. Data Nucl. Data Tables 14 177
Sarsa A, Gálvez F J and Buendı́a E 1999 J. Chem. Phys. 110 5721
Boys S F and Handy N C 1969 Proc. R. Soc. A 310 43
Schmidt K E and Moskowitz J W 1990 J. Chem. Phys. 93 4172
Linderberg J and Shull H 1960 J. Mol. Spectrosc. 5 1
Alexander S A and Coldwell R L 1995 J. Chem. Phys. 103 2572
Sarsa A, Gálvez F J and Buendı́a E 1998 J. Chem. Phys. 109 7075
Sarsa A, Gálvez F J and E. Buendı́a E 1998 J. Chem. Phys. 109 3346
Langfelder P, Rothstein S M and Vrbik J 1997 J. Chem. Phys. 107 8525