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Low- and high-momentum density functions in many-electron atoms
Low- and high-momentum density functions in many-electron atoms Toshikatsu Koga
*
Department of Applied Chemistry, Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan Received 16 May 2005; in final form 13 June 2005 Available online 1 July 2005
Abstract When any two electrons are considered simultaneously, the radial momentum density function I(p) in many-electron systems is shown to be rigorously separated into low-momentum I<(p) and high-momentum I>(p) density functions. Accordingly, radial momentum properties such as the Compton peak height J(0), the electronic kinetic energy T, and the mass–velocity correction energy emv are the sum of low- and high-momentum contributions. In relation to the shell structure of atoms, different characteristics of local maxima in the momentum densities I<(p), I>(p), and I(p) are reported for the 102 atoms He through Lr in their ground states. 2005 Elsevier B.V. All rights reserved.
1. Introduction The radial momentum density function, I(p), is defined (see, e.g., [1]) by Z 2 IðpÞ ¼ p dX PðpÞ; ð1aÞ where (p,X) are polar coordinates of the momentum p and Z 2 PðpÞ ¼ N dsdy 2 . . . dy N jUðy; y 2 ; . . . ; y N Þj ð1bÞ is the single-electron momentum density associated with a normalized N-electron momentum wavefunction U(y1,. . .,yN) with yi = (pi, si) being the combined momentum-spin coordinates of the electron i. The momentum density I(p) represents the probability density function of finding an electron at a momentum magnitude between p and p + dp, and it is normalized to N, the number of electrons. It is important that the density I(p) is intimately related with experimental measurements: within the impulse approximation, the isotropic Compton profile J(q) is given [1] in terms of I(p) by *
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0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.06.048
J ðqÞ ¼
1 2
Z
1
dp p1 IðpÞ;
ð2Þ
jqj
which conversely implies IðpÞ ¼ 2p½dJ ðpÞ=dp.
ð3Þ
Though I(p) is a single-variable reduction of the momentum wavefunction U(y1,. . .,yN) with 4N variables, radial momentum properties which depend solely on the momentum magnitude p are completely determined by the knowledge of I(p). Typical examples are momentum moments Z 1 n hp i ¼ dp pn IðpÞ; ð4Þ 0
which are related with several physical properties: Æp1æ is twice the peak height J(0) of the Compton profile J(q), Æpæ/p is close [2] to the Slater–Dirac exchange energy, Æp2æ is twice the non-relativistic electronic kinetic energy T, and Æp3æ is approximately proportional [2,3] to the initial value of the Patterson function in X-ray crystallography. The fourth moment Æp4æ determines [4] the Breit–Pauli mass–velocity correction energy emv, though its usefulness is criticized [5,6]. Another important aspect of the momentum density I(p) of atoms is that it exhibits (see, e.g., [7–11]) several local maxima
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T. Koga / Chemical Physics Letters 411 (2005) 243–247
corresponding to the electron shells. These maxima are also known [12] to be connected to the relative sizes [13,14] of atoms. The momentum density I(p) has been derived by focusing only on a single electron among N electrons. In the present Letter, however, we point out that if any two electrons are considered simultaneously, the momentum density I(p) is rigorously partitioned into two component functions: i.e., the low-momentum I<(p) and high-momentum I>(p) density functions. The low-momentum density function I<(p) represents the probability density that an electron of our attention has a momentum p which is smaller than that of the other electron, and the high-momentum density function I>(p) the probability density in the opposite situation. The partitioning of I(p) into the two components results in the corresponding partitioning of any radial momentum properties, including the moments Æpnæ. For the 102 atoms, He through Lr, in their ground states, the three momentum densities I<(p), I>(p), and I(p) are examined and the differences in their maximum characteristics are clarified based on the numerical Hartree–Fock wavefunctions. Relative contributions of the low- and highmomentum components to Æpnæ are also discussed for these atoms. Hartree atomic units are used throughout.
2. Low- and high-momentum density functions We first introduce the two-electron radial momentum density function I2 (p1, p2) given (see, e.g., [15,16]) by Z 2 2 I 2 ðp1 ; p2 Þ ¼ p1 p2 dX1 dX2 Cðp1 ; p2 Þ; ð5aÞ where (pi, Xi) are polar coordinates of pi and Z N ðN 1Þ Cðp1 ; p2 Þ ¼ ds1 ds2 dy 3 . . . dy N jUðy 1 ; . . . ; y N Þj2 2 ð5bÞ is the spinless two-electron momentum density function normalized to the number of electron pairs, N(N1)/2. The function I2 (p1, p2) is the probability density that one electron has the momentum magnitude p1 and the other electron the momentum magnitude p2, when any two electrons are considered simultaneously, and is normalized as Z 1 Z 1 N ðN 1Þ . ð6Þ dp1 dp2 I 2 ðp1 ; p2 Þ ¼ 2 0 0 Then the single-electron momentum density I(p), defined by Eq. (1a), is alternatively obtained from I2 (p1, p2) as Z 1 2 IðpÞ ¼ dp2 I 2 ðp; p2 Þ. ð7Þ N 1 0 We now insert into the integrand of Eq. (7) an identity relation
ð8aÞ H ðp p2 Þ þ H ðp2 pÞ ¼ 1 of the Heaviside function [17] H(x a) defined by 8 > < 0 x < a; ð8bÞ H ðx aÞ ¼ 12 x ¼ a; > : 1 x > a. We then have IðpÞ ¼ I < ðpÞ þ I > ðpÞ;
ð9Þ
where
Z 1 2 dp2 H ðp2 pÞI 2 ðp; p2 Þ N 1 0 Z 1 2 ¼ dp2 I 2 ðp; p2 Þ; ð10aÞ N 1 p Z 1 2 I > ðpÞ ¼ dp2 H ðp p2 ÞI 2 ðp; p2 Þ N 1 0 Z p 2 ¼ dp2 I 2 ðp; p2 Þ. ð10bÞ N 1 0 The function I<(p) is called the low-momentum function, since it represents the probability density that an electron of our attention moves with a momentum p which is smaller than the momentum p2 of the other electron. The function I>(p) is the probability density in the opposite situation and is called the high-momentum function. The low- and high-momentum density functions are normalized as Z 1 Z 1 N ð11Þ dp I < ðpÞ ¼ dp I > ðpÞ ¼ ; 2 0 0 I < ðpÞ ¼
in harmony with the normalization of I(p) to N. We see from the comparison of Eqs. (7), (10a), and (10b) that I<(p) is a dominant component function of I(p) when p is small while I>(p) is dominant when p is large. We therefore expect that I<(p) is sensitive to the valence or outer electrons which have lower momenta, whereas I>(p) is sensitive to the core or inner electrons which have higher momenta. Since the momentum density I(p) is rigorously separated into the low- and high-momentum densities, any radial momentum properties f(p) that depend only on the momentum magnitude p are partitioned correspondingly. As a simple example, for the momentum moments f(p) = pn we have hpn i ¼ hpn< i þ hpn> i; ð12aÞ where Æpnæ is defined by Eq. (4), and Z 1 hpn< i ¼ dp pn I < ðpÞ 0 Z 1 Z 1 1 ¼ dp1 dp2 pn< I 2 ðp1 ; p2 Þ; N 1 0 0 Z 1 hpn> i ¼ dp pn I > ðpÞ 0 Z 1 Z 1 1 ¼ dp1 dp2 pn> I 2 ðp1 ; p2 Þ; N 1 0 0
ð12bÞ
ð12cÞ
T. Koga / Chemical Physics Letters 411 (2005) 243–247
in which p< = min (p1, p2) and p> = max (p1, p2). A few important cases of Eq. (12a) are the following: When n = 1, we obtain a partitioning of the Compton peak height J(0). When n = 1, we have the average low Æp<æ and high Æp>æ momenta studied in [16] under the condition that all the momentum densities are normalized to unity. When n = 2 and n = 4, respectively, Eq. (12a) shows that the electronic kinetic energy T = Æp2æ/2 and the mass–velocity correction energy emv = Æ p4 æ /(8c2) are the sums of the low- and high-momentum contributions, where c is the speed of light.
245
Figs. 1 and 2 exemplify the momentum densities I<(p), I>(p), and I(p) for the six group-18 atoms. As mentioned in the previous section, we find in the figures that the low-momentum density I<(p) is a dominant component of I(p) for a small p, while the high-momentum density I>(p) is for a large p. Accordingly, the characteristics of local maxima are different among the three densities, though the largest number of local maxima is found in I(p) for many atoms. When we examine the low-momentum region corresponding to valence electrons, local maxima in I(p) originate mainly from I<(p). For example, the innermost maximum corresponding to the Q shell is observed for the 17 atoms Z = 87–103 in I<(p) and I(p), but for no atoms in I>(p). When we consider the high-momentum region corresponding to the core electrons, on the other hand, local maxima and shoulders in I(p) are mainly due to I>(p). Moreover, the latter density detects new maxima missing in I(p) for some atoms. For example, I>(p) shows the L-shell maxima for the atoms Z = 21 and 22
3. Results and discussion for atoms Using a modified version of the MCHF72 program [18], we have calculated the Hartree–Fock momentum densities I<(p), I>(p), and I(p) for the 102 atoms He (atomic number Z = 2) through Lr (Z = 103) in their ground states [19].
Momentum Densities
1.4 1.2
(a) He
I(p)
1.0 0.8
I<(p)
0.6 0.4 I>(p)
0.2 0.0 0.1
1
10
100
10
100
10
100
p
Momentum Densities
3.5 3.0
I(p)
(b) Ne
2.5 I<(p)
2.0 1.5 1.0
I>(p)
0.5 0.0 0.1
1 p
Momentum Densities
7.0 6.0
(c) Ar
I(p)
5.0 4.0
I<(p)
3.0 2.0 1.0
I>(p)
0.0 0.1
1 p
Fig. 1. Hartree–Fock momentum densities I<(p), I>(p), and I(p): (a) He atom with K-shell peaks, (b) Ne atom with L-shell peaks, (c) Ar atom with M-shell peaks. I>(p) and I(p) have L-shell peaks additionally.
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T. Koga / Chemical Physics Letters 411 (2005) 243–247
Fig. 2. Hartree–Fock momentum densities I<(p), I>(p), and I(p): (a) Kr atom with M- and N-shell peaks, (b) Xe atom with N-and O-shell peaks. I>(p) has an M-shell peak additionally, (c) Rn atom with N-, O-, and P-shell peaks.
and the M-shell maxima for the atoms Z = 48–58, where the corresponding maxima are not observed in I(p). For the 102 atoms examined, Table 1 summarizes how local maxima appear in the three momentum densities. The three kinds of momentum moments hpn< i; n hp> i; and hpn i have also been computed for n = 2 to +4. Among them, the calculated Æpnæ values have been verified by comparison with the literature values [20], except the Bk and Lr atoms for which different states were considered in [20]. For n = 2, 1, 1, and 2, the ratios hpn< i=hpn i of the low-momentum moments hpn< i relative to the total moments Æpnæ are depicted in Fig. 3 as a function of Z. When n < 0, the figure shows that the major parts of the moments come from the low-momentum density I<(p). In fact, the low-momentum contribution occupies 81.3% of the Compton peak height J(0) on average, with the minimum 67.9% at Z = 2 and the maximum 85.2% at Z = 88. Moreover, the ratios show a periodical Z-dependence reflecting the valence electron configuration of atoms. In particular, we have local
maxima for the group-1 or -2 atoms with diffuse s orbitals, whereas local minima for the group-18 atoms with fully-occupied orbitals. For transition metal atoms, we also find the difference between sdm and s2dm1 configurations for n = 2 and 1. When n > 0, on the other hand, the low-momentum contributions are much smal-
Table 1 Atomic numbers (Z) of ground-state atoms for which electron shells are detected as the local maxima in the three momentum densities Shell
I<(p)
I>(p)
I(p)
K L M N O P Q
2–4 3–15 11–42 19–91 37–103b 55–103 87–103
2–5 4–22 12–58 20–103a 49–103 84–93 None
2–5 3–20 11–47 19–99 37–103b 55–103 87–103
a b
Except for 24 and 29. Except for 46.
T. Koga / Chemical Physics Letters 411 (2005) 243–247
247
1.1 n = -2
1 0.9 0.8
n = -1
n
n
0.7 0.6 0.5 0.4 0.3
n=1
0.2 n=2
0.1 0 0
20
40
60
80
100
Z
Fig. 3. The ratios hpn< i=hpn i of the low-momentum moments hpn< i relative to the total moments hpn i for n = 2, 1, 1, and 2.
ler than the high-momentum ones, and the ratios are almost constants (0.23 for n = 1 and 0.08 for n = 2) except for the first several atoms. On average, the highmomentum density I>(p) is responsible to 91.7% of the electronic kinetic energy T and to 99.5% of the mass– velocity correction energy emv.
4. Concluding remarks When any two electrons are considered simultaneously, the momentum density function I(p) in manyelectron systems is separated into the low-momentum I<(p) and high-momentum I>(p) densities. Accordingly, radial momentum properties such as the momentum moments Æpnæ are the sum of low- and high-momentum contributions. Numerical results have been discussed for the 102 atoms He through Lr in their ground states. All the present theoretical results are valid both at the relativistic and non-relativistic levels and both at the Hartree–Fock and correlated levels. Though this Letter has presented numerical results from non-relativistic, Hartree–Fock calculations, we wish to study in the future how the relativistic and/or correlation effects influence the low- and high-momentum density functions.
Acknowledgements The author thanks Mr. T. Shimazaki for his computational assistance. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry
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