Accurate Roothaan-Hartree-Fock momentum expectation values for ground states of the atoms He to Xe

28 April 1995

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 236 (1995) 616-620

Accurate Roothaan-Hartree-Fock momentum expectation values for ground states of the atoms He to Xe J.M. Garcia de la Vega, B. Miguel Departamento

de Quimica

Fkica

Aplicada,

Facultad

de Ciencias,

Universidad

Authoma

de Madrid,

28049 Madrid,

Spain

Received6 December 1994;in final form 14 February 1995

Abstract Two new sets of RHF wavefunctions with more accurate energies for the ground states of atoms up to Z = 54 have been published. We report momentum space properties calculated from these new RHF wavefunctions and compare them with those obtained previously for the atoms He to Xe . The behaviour and validity of the powers of (p”) through the periodic system are analyzed in order to select adequate RHF wavefunctions in momentum space.

1. Introduction In recent papers [ l-31, we tabulated the momentum expectation values of the ground and excited states of elements up to 2 = 54, employing the RoothaanHartree-Fock (RHF) wavefunctions of Clementi and Roetti [ 41 (CR), for 2 = 55 to 2 = 92 using the RHF wavefunctions of McLean and McLean [5], and for 2 = 92- 103 using the triple-zeta-valence (extended) Hartree-Fock-Slater wavefunctions of Snijders et al. [6]. Additionally, two new sets of RHF wavefunctions have been published for the ground states of the atoms He-Xe by Bunge, Barrientos, Bunge and Cogordan [ 7,8] (BBBC) and Koga, Tatewaki and Thakkar [ 9,101 (KTT) . These wavefunctions are expanded in Slater type orbitals (STOs) using Roothaan’s procedure [ 111. These wavefunctions, whose atomic energies are close [7,9] to numerical Hartree-Fock (NHF) ones [ 121, are widely applicable in atomic physics and quantum chemistry. The KTT basis sets have the same number of functions as the CR ones [ 41, while the BBBC basis sets are

larger and consequently give energies closer to the NHF results. The features of RHF wavefunctions are important because molecular calculations are carried out with a stationary nucleus when a nonrelativistic Hamiltonian is used. The electronic density distribution of atoms in momentum space plays an important role in the atomic properties of real atoms. The increasing use of (e, 2e) spectroscopy, Compton scattering and positron annihilation [ 131 as probes for the study of electron momentum distributions has made it desirable to obtain accurate momentum densities and Compton profiles. For comparison with experimental data, a theoretical total Compton profile for the free atom is needed. Some of the most interesting applications of Compton scattering refer to the determination of the electronic properties of solids. To analyze experiments on solids correctly, a theoretical atomic-core profile subtraction must be made in order to study the electronic properties of the valence and conduction electrons. It now appears likely that scattering effects may make possible the measurement of several of the expectation

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J.M.

Garcia

de la Vega, B. Miguel/Chemical

values 07”). For instance, the powers of atomic moments are directly related to some properties and show a strongly periodic variation [ 141. (p-t)/2 is the peak height of the Compton profile J( 0) [ 151. An approximation to the Slater-Dirac exchange energy of density functional theory is given by 7r (p’) [ 161 and (p’) is also correlated with the shielding in nuclear magnetic resonance spectroscopy. The electronic kinetic energy is b2/2). (p3) is proportional to the initial value of the Patterson function in X-ray crystallography [ 171. In the BreitPauli approximation, the energy due to the relativistic variation of mass velocity [ 181 is -((y2/8)(p4), where (Y is the fine-structure constant. The electron correlation and relativistic effects contribute to the momentum space properties, although a complete and systematic study has not been carried out to date. Tripathi et al. [ 191 have examined the importance of electron correlation on momentum densities and other density related properties using available correlated wavefunctions. They report the study of the accurate momentum density, Compton profile, and other quantities related to the momentum density , for Be and its isoelectronic sequence using CI wavefunctions. The results reveal differences that indicate that the inclusion of electron correlation is necessary if one is to obtain highly accurate momentum space quantities. The present calculation clearly demonstrates that the effects of the electron correlation are important in obtaining accuracy, particularly in the valence region. In the construction of accurate densities, the density convergence criteria are superior to those based solely on the correlation energy. In a previous paper [ 21, we have compared (p-r) values for the Cs and U atoms calculated with relativistic and nonrelativistic wavefunctions. Our results show that the relative error between the relativistic and nonrelativistic values increases with atomic number. The total values are greater than the relativistic ones. The orbital values confirm the well-known behaviour of relativistic calculations, namely the energetic stabilization of inner orbitals and the energetic destabilization of outer orbitals. Due to the wide use of STOs at the RHF level in physics and chemistry, the systematic tabulation of momentum expectation values for atoms is worthwhile. A detailed description of the analytic expressions for the orbital and total momentum values of

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617

atoms up to Z = 92 using wavefunctions expanded in STOs is summarized elsewhere [ 2,201. In this Letter, we compute powers of momentum expectation values with these new basis sets (BBBC and KIT) and analyze the results obtained, in order to select the most appropriate RHF wavefunctions that give the best expectation values.

2. Results and discussion

The accuracy of momenta is limited by the number of figures in the exponents and the coefficients of each basis set [ l-3,20]. Thus, in order to obtain more accurate results, the orbital coefficients of the BBBC and KIT basis sets have been recalculated to 8 significant figures, using the ATOMSCF program included in the MOIECC package [ 211. The use of coefficients from the original tables [ 8,101, with only 6 figures in fixed format, gives poor accuracy in the calculation of (p”) values. Insofar as the number of significant figures for exponents and coefficients is 6 (or 7) and 8 for KIT wavefunctions and 4 (or 5) and 8 for BBBC wavefunctions, we can conclude that 8 and 6 significant figures are valid for (p”) values using KTT and BBBC wavefunctions, respectively. However, our results are expressed with 8 figures for both wavefunctions, taking into account that the last 2 figures of (p”) values with BBBC wavefunctions can be improved using more accurate exponents. The results of (p”) values given in atomic units have been computed for the ground state of each atom using BBBC and KTT wavefunctions. In order to compare these new results with those previously calculated with CR wavefunctions [ 11, we report a comparative summary for (p-*) in Table 1 and for (p2) in Table 2 for closed shell (alkaline-earth) and open shell (nitrogen group) atoms. Table 1 summarizes the 6-l) values calculated with the CR, KTT and BBBC basis sets and values obtained by Biggs et al. [ 221 (for He to Kr), and by Benesch [23] (for As to Yb) with NHF wavefunctions. Unfortunately, the data of Biggs et al. are presented with only 3 or 4 digits; due to this low accuracy it is only possible to compare the order of magnitude of (p-l) expectation values for the He to Sr atoms. Taking into account the rows of the periodic table, the highest values using RI-W and NHF wavefunctions,

618

J.M.

Table 1 (P-I) values Atom

Garcia

for atoms with RHF CRa

de la Vega, B. Miguel/Chemical

BBBC

6.3182392 5.5975330 10.293651 10.134746 15.748991 14.100847 20.490258 19.226361

a Ref. [l]. b Ref. [ 22 1, except

6.3185645 5.5969383 10.299877 10.137358 15.733563 14.106124 20.426397 19.168566

6.3177016 5.5972547 10.309409 10.141027 15.733862 14.106962 20.493601 19.224843

6.32 5.66 10.32 10.22 15.74 14.1079 20.498 1 19.2275

for the As, Sr and Sb atoms from Ref. [ 231.

Table 2 (p*) values

for atoms with RHF

Atom

CRa

KTT

BBBC

NHFb

Be( ‘S) N(4s) W’S) V4S) Ca( IS) AS Sr(‘S) Sb(4S)

29.146072 108.80104 399.22050 681.42762 1353.5125 4468.4749 6262.9530 12626.783

29.146043 108.80186 399.22925 681.43755 1353.5162 4468.4770 6263.0835 12626.963

29.146048 108.80187 399.22928 681.43755 1353.5164 4468.4769 6263.0913 12626.971

29.146046 108.80186 399.22918 681.43756 1353.5163 4468.4770 6263.0914 12626.97 1

a Ref.

[l].

b Ref.

236 (1995)

616-620

NHFb A&BC

Be( ‘S) NC49 W’S) p(4s) Ca( ‘S) AS Sr(‘S) Sb(4S)

Letters

and the previous values of (p”) from Ref. [ I]. We define Ag,nc and Ak-rr as

wavefunctions

KTT

Physics

= WBBBC

Ak.,-r =

(pn)KTT

-

W)CR,

(1)

(2)

(/?)CR.

Figs. 1 to 3 show the differences in atomic momentum expectation values for the new wavefunctions with respect to the CR wavefunctions (AEano and Akrr) for the atoms up to 2 = 54. Note that different scale factors are applied and that the shapes of each pair of

A-’

wavefunctions

-0.1 -0.2

-

-0.3

-

-0.4

-

-0.5

-

-0.6

-

0.7 -0.6

-

-0.9

-

-1.0

-

-1.1

-

-1.2

-

[12].

are obtained from alkaline-earth atoms. The comparison of values between RHF and NHF wavefunctions is accurate except for the C, N, Si, P, Cr, Ge, As and Se atoms, in which less than 4 digits are obtained. In those atoms the expectation values obtained with NHF wavefunctions are always higher than those obtained with RHF ones. The results of (p’) are summarized in Table 2 where the close agreement between the NHF, K’IT and BBBC results is shown. Only for Sr and Sb (both belonging to the Rb-Xe period) do the KIT results have less accuracy. For some atoms, the BBBC, KIT and NHF wavefunctions produce results equal to 7 or 8 significant figures. The values with BBBC wavefunctions are greater than those with K’lT for Rb-Xe atoms, whereas the values with both wavefunctions are similar for the other atoms. We have tested (p”) (n = -2 to 4) for their sensitivity towards changes in the wavefunction quality. Here we present the difference between the corresponding value of (p”) using BBBC and K’IT wavefunctions

60

z

A-’

lb)

0.05 0.04 0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.06

-0.09

6

! 0

20

40

60

Z Fig. 1. (a) (pm*) differences A;&; (0) A&. (b) (p-r) number.

(+)

A&&c;

(0)

(in au) versus atomic number. (+) differences (in au) versus atomic

A$,..

J.M.

Garcia

de la Vega, B. Miguel/Chemical

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236 (1995)

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616-620

n”

(a)

160 150 140 130 120 110 100 so 80 70 60 50 40 30 20 10 0 -10 0

M

40

60

Z

Z

@I

(b) 100

2.1 2.0 1.9 1.6 1.7 1.6 1.5 i.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

t so 60 70 60 zi P E

50 40 30 20 10 0

0

20

40

6C

60

2

Z fig. 2. (a) (PI) differences (in au) versus atomic number. (+) AkeBc; (0) Al ~. (b) (p2) differences (in au) versus atomic 5 number. (+) ABBBC; (0) A&

fig. 3. (a) (p3} differences (in au) versus atomic number. (+) A&,Bc; number. (0) (+) A3 Aizc.C~;f~~ difimmces (in au) versus amnic

figures corresponding to n = -2 and - 1 (Fig. 1) , to IZ = 1 and 2 (Fig. 2) and to n = 3 and 4 (Fig. 3) are similar. The differences for negative powers (n = -2 and - 1) show a similar relation (see Fig. 1). Thus, they show positive differences for A$nc and A&& and . . negative differences for Ai& and A& from the Rb atom (Z = 37). Appreciable differences are produced in the negative powers (p-‘) and (p-l), where the BBBC and KTT wavefunctions have opposite variations from the CR momentum expectation values. Fig.

2 shows the differences for (p’) and (p2). The differences are negligible, except for the heavier atoms, from the Rb atom. The maximum corresponds to the Cd atom (Z = 48) for both A’ and A2 differences, which is related to the best optimization in the energy using the BBBC and K’IT wavefunctions [ 7,9]. (p’ ) and b’) are proportional to the Slater-Dirac exchange energy and electronic kinetic energy and so both values are improved with the employment of the new RHF wavefuntions. The differences corresponding to the higher powers (p3) and (p4) are depicted

4 Tr'

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in Fig. 3, being close to zero for light atoms (up Z = 30) and positive for heavier atoms, specially from Rb atom (Z = 37). These differences increase for both the BBBC and KIT wavefunctions, with the exception of the Cd atom, which is related with the optimization of the CR basis sets [ 7,9]. The momentum expectation values with Z ,< 54 calculated from the new RHF wavefunctions (BBBC and KIT) may be employed to study the quality of RHF wavefunctions due to its direct relation with atomic physical properties. Although the BBBC and K’IT wavefunctions improve from 7 or 8 significant figures the energy calculated with the CR ones, we have found that the calculated theoretical values change especially (pe2) and (p-i) for heavy elements, and that the Compton profiles so obtained may be different. Differences in the power of the momenta vary with atomic number. For Rb through Xe, the BBBC wavefunctions are more accurate than K’IT wavefunctions. Using these two new basis sets we obtain values of momenta for those atoms which are more different from those obtained for all other atoms. A complete tabulation of the total and orbital momentum expectation values and recalculated RI-IF coefficients are available from the authors upon request. Acknowledgement

This work was financially supported by the Direcci6n General de Investigation Cientifica y Tt%nica (DGICYT) of Spain (grants PB91-0010 and PR94084). The authors thank Professor A.J. Thakkar for making available a copy of his RI-IF wavefunctions. References [ l] J.M. Garcia de la Vega and B. Miguel, At. Data Nucl. Data Tables 54 ( 1993) 1.

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121J.M. Garcia de la Vega and B. Miguel, At. Data Nucl. Data Tables 58 (1994) 307. ]31 J.M. Garcia de la Vega and B. Miguel, At. Data Nucl. Data Tables, to be published. [41 E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [51 A.D. McLean and R.S. McLean, At. Data Nucl. Data Tables 26 (1981) 197. 161J.G. Snijders, P Vemooijs and E.J. Baerends, At. Data Nucl. Dam Tables 26 (1981) 483. [71 CF. Bunge, J.A. Barrientos, A.V. Bunge and J.A. Cogordan, Phys. Rev. A 46 (1992) 3691. [81 CF. Bunge, J.A. Barrientos and A.V. Bunge, At. Dam Nucl. Data Tables 53 (1993) 113. 191 T. Koga, H. Tatewaki and A.J. Thakkar, Phys. Rev. A 47 (1993) 4510. I101 A.J. Thakkar, private communication. 1111C.C.J. Roothann, Rev. Mod. Phys. 23 (1951) 69; 32 ( 1960) 179. 1121C. Froese-Fischer, The Hartree-Fock method for atoms (Wiley, New York, 1977). 1131 W.A. Reed, Acta Cryst. 32A ( 1976) 676. [I41 A.J. Thakkar and T. Koga, Intern. J. Quantum Chem. 26s (1992) 291. [I51 B.G. Williams, ed., Compton scattering: the investigation of electron momentum distributions (McGraw-Hill, New York, 1977). [ 161 R.G. Parr and W. Yang, Density functional theory of atoms and molecules (Oxford Univ. Press, Oxford, 1989). [17] J.P Glusker, B.K. Patterson and M. Rossi, eds., Patterson and Pattersons (Oxford Univ. Press, Oxford, 1986). [ 181 S. Fraga, M. Klobukowski, J. Muszynska, E. San Fabian, K.M.S. Saxena, J.A. Sordo and T.L. Sordo, Research in atomic structure (Springer, Berlin, 1993). [ 191 A.N. Tripathi, RI? Sagar, R.O. Esquivel and V.H. Smith Jr., Phys. Rev. A 45 (1992) 4385. [20] J.M. Garcia de la Vega and B. Miguel, J. Math. Chem. 14 (1993) 219. [21] E. Clementi, S.J. Chakravorty, G. Corongiu and V. Sonnad, Modem techniques in computational chemistry: MOTECC90, ed. E. Clementi @scorn, Leiden, 1990) p. 47. 1221 E Biggs, L.B. Mendelsohn and J.B. Mann, At. Data Nucl. Data Tables 16 (1975) 201. [23] R. Benesch, Can. J. Phys. 54 (1975) 2155.