helium momentum-space wave function and compton profile

PHYSICAL REVIEW A VOLUME 38, NUMBER 7 OCTOBER 1, 1988

Helium momentum-space wave function and Compton profile

Paul J. SchreiberAir Force Wright Aeronautical Laboratories, Wright-Patterson Air Force Base, Ohio 45433-6543

R. P. HurstDepartment of Physics and Astronomy, State University ofNew York at Buffalo, Amherst, New York 14260

Thomas E. DuvallAeronautical Systems Division, Information Systems and Technology Center, Wright Patte-rson Air Force Base, Ohio 45433-6543

(Received 2 May 1988)

This paper describes an effort to develop further the work of R. McWeeny and C. A. Coulson[Proc. Phys. Soc. London, Sect. A 62, 509 (1949)] to obtain a helium-atom ground-state wave func-

tion in momentum space. In the application of Svartholm's method [N. V. Svartholm, Ph. D. thesis,University of Lund, Sweden (1945)] to the helium momentum-space integral Schrodinger equation,a Feynman-integral evaluation technique is used to start the iteration in the electron-electron in-

teraction term. The resulting correlated momentum-space wave function is used to evaluate thehelium-atom ground-state energy and Compton profile.

I. INTRODUCTION

Calder et al. ' have made an x-ray di6'raction analysisof LiH and from the experimental measurements of thescattered x rays have evaluated LiH structure factors,atomic-scattering form factors, and electron distribu-tions. They have compared the experimental andtheoretical values of these physical quantities. Theirtheoretical predictions are based on LiH coordinate-space atomic wave functions calculated by a number ofauthors. It has been determined by comparison that,along with others, a Hurst crystal-field calculation, inwhich the neighbors are assumed to be point charges,very well represents LiH. Phillips and Weiss have calcu-lated and measured the LiH Compton profile. They usemomentum-space wave functions in their profile calcula-tions which are the Fourier transform of the Hurst LiHopen- and closed-configuration wave functions. Theycompared theoretical and experimental LiH Comptonprofiles and found them not to be in agreement. OtherLiH theoretical and experimental Compton profile com-parisons have also been made. '

A possible step toward a more accurate LiHmomentum-space wave function might be found in calcu-lation of the LiH wave function directly in momentumspace. Lithium hydride is structured as sodium chloridewith a basis of Li+ and H ions. These ions are helium-like with two electrons per ion. If an approximate solu-tion to heliumlike atomic systems could be realized inmomentum space, an attempt to model LiH in momen-tum space might succeed as well.

Heliumlike atomic systems in momentum space havebeen considered previously which seek a momentum-space ground-state wave function and energy. McWeenyand Coulson Fourier transformed a heliumlikecoordinate-space Schrodinger equation into momentumspace in which the electron-electron electrostatic interac-

tion energy was of Coulomb form. They applied thevariation-iteration method of Svartholm to themomentum-space Schrodinger equation in which thestarting function was chosen to be the momentum-spaceform of the product of two hydrogen s-type wave func-tions. These authors found an upper bound to theground-state energy as Eo= —(Z —

—,', ) a.u. , but did notobtain a first-iteration wave function for helium inmomentum space.

Henderson and Scherr, as well as many others,have also considered heliumlike atomic systems inmomentum space in which the electron-electron electro-static interaction energy was taken to be of Coulombform in the coordinate-space Schrodinger equation.Ground-state energies have been found in excellent agree-ment with experiment.

In this paper, our approach to heliumlike atomic sys-terns in rnomenturn space is the same as taken byMcWeeny and Coulson but with one new and significantstep: we use a Feynman-integral evaluation technique toperform the iteration of the product of two hydrogen s-

type momentum-space wave functions in the integralSchrodinger equation. We calculate a first-iterationmomentum-space wave function in which the electron-electron interaction part of the wave function is evalu-ated both analytically and by Gaussian quadrature. Thisnew momentum-space wave function has correlationthrough the factor (p&+p2) as a result of the iteration.We examine the accuracy of the ground-state wave func-tion by evaluation of the ground-state energy usingSvartholm's method. We also evaluate the average valueof the electron's kinetic energy and then use the virialtheorem to determine the value of the average potentialenergy of the electrons. This average value for the totalenergy of the electrons determined by the use of the virialtheorem is compared to the calculated energy and exactenergy. The helium-atom Compton profile is evaluated

38 3200 1988 The American Physical Society

38 HELIUM MOMENTUM-SPACE WAVE FUNCTION AND COMPTON 3201

and compared to the measured proSe.The contents are divided into sections as follows. In

Sec. II, we briefly review the helium integral Schrodingerwave equation in momentum space and the Svartholmvariation-iteration method in order to establish our nota-tion. In Sec. III, we calculate the form of the electron-electron interaction integral which is part of the wavefunction. In Sec. IV, we evaluate by Gaussian quadraturethe electron-electron interaction integrals. In Sec. V, weevaluate the value of the variational parameter and theenergy which then become numerical constants in thewave function; we then compare the calculated heliumground-state energy to its experimental value. In Sec. VI,we use the first-iteration wave function to evaluate thehelium Compton profile and kinetic energy. We then ap-ply the virial theorem to determine the average ground-state energy which is compared to the value of the energydetermined from the application of Svartholm's method.In Sec. VII, we summarize our results and state our con-clusions. In the Appendix, we evaluate analytically theelectron-electron interaction integrals.

II. WAVE EQUATION, METHOD OF SVARTHOLM,AND SYMMETRY OF WAVE FUNCTION

I&((t ) = fd p p P(pt —p, pz),

Iz(P)= f dPP P(P&, pz —P),

Ilz(p)= f dPP (t'(P& —P, Pz+P),

where

pp ———2E,2=

and

(2.2)

(2.3)

(2.4)

(2.5)

A, = 1/m (2.6)

The method of Svartholm' as applied to heliumlikesystems first requires the evaluation of the followingquantities when r =0:

(PO+P I +P2 ) '[ZIi(p")+ZIz($") —Ilz(4'")]

(2.7)

f0 (PO+P1 +Pz)4 dpldpz

T„=fp"(po+p i +pz )p"d p&dpz,

(2.8)

(2.9)

(2.10)

and

+ [ & y2 ]= W& / T& + i (2.11)

The nonrelativistic Schrodinger equation in momen-tum space for a system of two electrons about an infinitenuclear mass has been given previously in atomic units.With Z equal to the nuclear charge, the wave equation is

(PO+p1 +Pz)4(pl Pz)

=A [ZI 1 ((() )+ ZIz(P) —I,z($)], (2.1)

where

III. FIRST-ITERATION WAVE-FUNCTION FORM

The first-iteration helium ground-state wave functionP'(p, , pz) is gimme~ by Eq. (2.7) when r =0. Previously,McWeeny and Coulson evaluated I&($0) and Iz($0), andso we only have to evaluate I,z(P ) to determineP'(p&, pz). We have

~ /a(p', +a')(p'z+a')'

2/I(P)=(p', +a')'(pz+a')

I, ($ )= (3.1)

(3.2)

The first-iteration wave function (FIWF) can be obtainedfrom (2.7) after evaluation of I&z(P ) in (2.4) since Ii(P )

and Iz($ ) have already been evaluated for helium. TheFIWF is then used to directly evaluate Wo of (2.8). Thisdirectly calculated Wp can be compared to the value ofWp given by the analytical formula for Wp obtained byMcWeeny and Coulson. They obtained their form ofWp without the use of the explicit form for the FIWF.Thus the comparison of Wp obtained in these twodifferent ways can be made for any value of the variation-al parameter except zero and is used as a criterion of ac-ceptance for the FIWF. If Wp obtained by the explicituse of the FIWF in (2.8) does not agree with the value ofWp given by the McWeeny and Coulson analytical Wp,then something is wrong with the FIWF obtained from(2.7).

Finally, when the two values of Wp agree, the calcula-tion of T, is in order, using (2.9) where E= —2. 8477 a.u.as determined from A,p by McWeeny and Coulson. AfterT, is evaluated, A, , &2 is determined using Wp and T, in(2.11). The numerical value of A, , zz is varied as a functionof the variational parameter. At the minimum value ofk»2, the value of E in A,

& i2 is changed fromEo= —2. 8477 a.u. until A, , iz ——1 /m in accordance with(2.6). Thus the values of a, zz and E,zz have been found,and the procedure is repeated for A,

After Wi has been calculated using (2.8), then W& and

T, are used in (2.10) to determine A, The minimumvalue of A,

&is found by varying the variational parameter.

At the minimum value of A,&

the value of E is changedfrom Eo ———2. 8477 a.u. until kl ——lie Thus the. valuesof a, and E„which must be used in the first-iterationwave function P'(p, , pz), have been determined. It is thisP'(p, , pz) which is used to calculate the Compton profileand kinetic energy.

The symmetry of the closed- and open-configurationtota1 wave function for the two-electron atomic systemhas to be in accordance with the exclusion principle.Thus the spin of the two-electron system has to be con-sidered. The ground-state total wave function has to beantisymmetric; consequently, that requires a symmetricmomentum space and antisymmetric spin-zero functionfor the closed-configuration (one variational parameter)total wave function. The open-configuration (two varia-tional parameters) total wave function is the product of asymmetric momentum space and antisymmetric spin-zerofunction.

3202 PAUL J. SCHREIBER, R. P. HURST, AND THOMAS E. DUVALL 38

and the initial function

(p2 + a 2)2(p2 +u 2)2(3.3}

We substitute L, (u) of (3.12) into Lp(p) of (3.8) andthen substitute Lp(p) into I,2(p ) of (3.5) which, after in-terchanging the integration over u and p, gives

We begin evaluation of I&2(p ) by substituting p of(3.3) into I&2(p ) of (2.4),

I„(P')=2~f L,,(u)du,0 P3

with

(3.13)

I„(yP)=l p l' [(P& P)'+u ] [(P2+P) +u']'

(3.4)

L2(u)= f p(P' 2PP—3+si+$2)

We can then write I,2(p ) as

I 2(0')= fLp(p)dp

with

where

$1 (P 1 P2 )u2 2

dp

p p +2pp3+$1+$2(3.14)

(3.15)dQ

Lp(p) =p1 —p +a p2+p +a

(3.6) and

2 2$2 =P2+a (3.16)

and where we have used d p =p dpd 0 anddQ=sin8d ed/. Feynman' has given an identity whichleads to

1 & u(1 —u)du2 2=

afb, o [a,u+b, (1—u)](3.7)

In the first integral on the right-hand side of (3.14), welet p =x and in the second integral on the right-hand sidewe let p = —x'. Since x' is a dummy variable, the in-tegrals in (3.14) combine and L2(u) becomes

We apply Feynman's identity to (3.6) by making theidentification L2(u)= f—~ x(x —2xp3+s&+$2)

(3.17)

and

ui =[(pi —p)'+u']

bi =[(P2+P)'+u'] .

L2(u) can be evaluated with standard integrals, ' ofwhich one can have three possible forms dependent onthe discriminant 5 given by

L ( 1)=uf(q —2p p3)'

with

(3.9)

With these a, and b„ the use of the identity (3.7) allows

Lp(p) of (3.6) to be written as

Lp(p)=6f u(1 —u)L, (u)du, (3.8)0

where

6 =4apcp —b p2

From L2(u) of (3.17) we have in (3.18)

a0 $1+$2

bp = —2p

CO=1,

so that 5 becomes

(3.18)

(3.19)

and

q =p'+(8 —p2) +up2+ u

p3 p1u p2( I

(3.10)

(3.11}

6=4[p4( —u +u)+a ],where

P4 =(Pi+P2) ~

2 2

(3.20)

(3.21)

In order to evaluate the integral L, (u) of (3.9), we letthe z axis in L, (u) lie along p3 and let x =cos0. We in-

tegrate over P and use a standard integral' with the re-sult

We can see in (3.20) that b, )0 for all u such that0& u & 1. We may now evaluate L2(u) of (3.17) with theresult

L, (u)=2nf.—

& (q —2pp3x)

2K 1 1

6pp3 (q 2PP3) (q +2pP3 —}(3.12)

L2(u) =L3(u)+L4(u)+Ls(u),

where

3&P3L3(u}=8(s, +s2)[p4( —u +u)+a ]" '+

(3.22)

(3.23)

where p =l p l, p3 =

l p, u —p2(1 —u) l, and q is given by(3.10).

7TP 3L4(u)=2(s +s ) [p ( —u +u)+a ]" (3.24)

38 HELIUM MOMENTUM-SPACE %'AVE FUNCTION AND COMPTON. . . 3203

and We substitute L2(u) of (3.22) into I)2(p ) of (3.13) andobtain

7TP 3L, (u)=($+@)3[p2(u2+u)+g2]1/2

(3.25)I)2((I) )=2m. (L6+.L2+Ls),

where

(3.26)

and

1 (u —u )du

Q[(p2p2)u+p2 +g2][p2(u2 +u)+g2](1 /2)+2

1 (u —u )du

() [( 2 2)u + 2+g2]2[ 2( u2+ )+ 2](1/2)+1~

1 (u —u )duLs= ' [(p) —p2}u+p2+g']'[p4( —u'+u}+g']'"

(3.27)

(3.28)

(3.29)

We obtain the first-iteration momentum-space wave function from (2.7), (3.1), (3.2), and (3.26), which give

1 Zn. /a Zm /a

(p()+p)+p2} (p)+a )(p2+a ) (p, +a ) (p2+a )(3.30)

where Lz, L7, and Ls are given, respectively, by (3.27),(3.28), and (3.29). The next step is to evaluate the in-tegrals L6, L7, and L, .

IV. EVALUATION OF THE ELECTRON-ELECTRONINTEGRALS

The evaluation of the electron-electron interaction in-tegrals given by (3.27)—(3.29) can be performed both nu-merically and analytically. We have evaluated these in-tegrals both ways and compared their values for a rangeof values of p1 and p2. This comparison has helped to es-

tablish an acceptable analytical representation for each ofthe interaction integrals. Even though we have evaluatedthese integrals analytically, it is more convenient to usetheir numerically evaluated forms when making calcula-tions using the FIWF. Consequently, we defer until theAppendix the analytical evaluation of L6, L7, and Ls.

We have numerically evaluated each of the electron-electron interaction integrals by an 11-term Gaussianquadrature. In order to use Gaussian quadrature tables,we have changed the variable of integration in (3.27),(3.28), and (3.29) by the relationship t =2u —1. We ob-tain

I -,'(t +1}—[-,'(t +1)]'j dt

5 t+1 +~2 +4 t+1 2+ I t+1 +a2 (1/2)+2

11= —,', g A;f6(t; ),

(4.1)

(4.2)

I-,'(t +1)—[-,'(t +1)]'jdt1 X5 21 t+1 +X2 X4 —

21 t+1 +21 t+1 +a' ""'+11

=-,' g A,f,(t, ),

(4.3)

(4.4)

+1 [ —,'(t +1)—[—,'(t +1)]'jdt

[x5[—,'(t+1)]+x2j (x4[ —[—,'(t+1)] +,'(t+1)j+g )'11

= —,' g A,f, (t, ),i=1

(4.5)

(4.6)


where f6(t, ), f7(t, ), and fs(t; ) are the integrands of (4.1),(4.3), and (4.5), respectively, and where A; and t, aregiven in a Gaussian quadrature table. ' In these integralswe have set Wo (calculated)

Wo (exact)2 2x1 ——p1+a2 2

x2 ——p2+a2 2 2 2

X3 Pp+P1+P2 Pp= —2E,

X4 J 1+P2+2P1 P2 &

2 2

2 2XS =11 P2 ~

We now can write the form of the FIWF as

0.100 768 40.100 769 8

0.100 757 80.100 759 1

0.100749 70.100 751 1

0.100 744 20.100 745 5

0.100 741 20.100 742 5

0.100 740 60.100 741 90.100 741 20.100 742 5

0.100 742 40.100743 70.100 759 50.100 760 80.101 652 40.101 653 8

1.475 493 01.475 51341.395 031 81.395 050 81.31941641.319434 51.248 327 61.248 344 61.181 469 41.181 485 21.118566 01.118581 01.088 516 81.088 531 21.059 361 81.059 375 90.913 740 90.913753 20.198037 30.198040 0

1.6000 14.642 409 8

13.845 396 5

13.095 977 0

12.391 058 7

11.727 766 4

11.103 425 2

10.805 076 7

10.515 546 3

9.068 533 4

1.948 1804

(4 7) 1.6100

1.6200

1.6300

1.6400

1.6500

1.6550

1.6600

1.6875

2.0000

Zn /a Z~ /a2 +

X1X2 X 1X2X3

11

2n ——,', g A;f6( t; )

11

+-,' g A,f, (t, )

11+i g A fs(t) (4.8)

TABLE I. Values of Wo Tl and A. , /2 as a function of thevariational parameter a for Z =2, and E = —2.8477 a.u. forhelium. The exact and computed values of Wo are shown.

where f6, f7, fs, A;, and t; have been defined previouslyIn Sec. V we will determine the variational parameterand the value of E in the FIWF using the method ofSvartholm.

V. EVALUATION OF THE VARIATIONALPARAMETER AND THE GROUND-STATE ENERGY

An objective of the method of Svartholm is to deter-mine the value of the variational parameter and the ener-

gy in the FIWF. This is achieved by minimization of thequantities A,p, A, 1i2, and k„however, an opportunity tocheck the first-iteration wave function is also possible inthe evaluation of Wp used in the evaluation of A,p andA, 1i2. McWeeny and Coulson previously evaluated Wpanalytically. They evaluated Wp by first integrating over

p, and pz in (2.8), and in this manner they were able toavoid the requirement to use the explicit form for P' inWp. Thus McWeeny and Coulson obtained an analyticalform for Wo without an explicit form for P in (2.8). Acomparison can then be made of the analytical value ofWp obtained by McWeeny and Coulson and of Wp ob-tained by using the FIWF of (4.8). This comparison ofWp values is shown in Table I. It shows that the Wp cal-culated with the FIWF compares favorably with the Wpdetermined by the analytical form of Wp and assures thatour FIWF of (4.8) is of the correct form.

McWeeny and Coulson evaluated A.p and determinedthat a value of the variational parameter ap = 1.6875 min-imizes A,p and that a value of Ep ———2.8477 a.u. allows Ap

to take on a value 1/m . The next step is to evaluate A, 1i2in (2.11) and then to minimize A, &i2 as a function of thevariational parameter. In the calculation and subsequentminimization, the value of E is taken to be Ep ———2. 8477

'An estimated relative error of 7)&10 ' was attained in thecomputation of T, . Thus the bottom value for A, &/, is also goodto a relative error of 7)& 10 ' (almost seven significant digits).

a.u. Table I shows A, 1i2 as a function of the variationalparameter. It can be seen that A, »2 is a minimum when

a, z2——1.65; with this value, A, , &2 is recomputed with a

new value of E. The value of E chosen is the one thatmakes A, , zz equal to 1/nIn thi. s. way we have found

E1i2 ———2.8805 a.u. for helium.The next step in the process is to evaluate and to mini-

mize A.&

which is given by (2.10) as I,&

——T&/W&. Theevaluation of W, requires the use of 1))' '(p, , pz) whichcan be obtained by substitution of P"'(p„pz) of (4.8) into(2.7). After laying p, along the z axis, W, is reduced to a

1.51.531.561.571.581.591.611.621.631.641.651.6875

26.106 834 021.868 525 418.376 896 617.353 581 216.392 782 5

15.490 344 5

13~ 845 396 5

13.095 977 012.391 058 711.727 766 411.103 425 29.068 533 4

259.320 091 0217.344 228 9182.709 477 5

172.553 267 3163.019035 5

154.046 656 6137.694 160 6130.249 522 1

123.238 686 2116.637 525 4110.421 147 090.153 892 7

0.100674 20.10061700.100 579 90.100 569 40.100 557 5

0.100 556 20.100 551 80.100 545 30.100 545 20.100 548 8

0.100 555 20.100 589 5

TABLE II. Values of T&, W&, and A, , as a function of thevariational parameter a for Z=2 and E= —2.8477 a.u. forhelium.

38 HELIUM MOMENTUM-SPACE WAVE FUNCTION AND COMPTON. . . 3205

TABLE III. Energies for three atomic systems using the method of Svartholm, virial theorem, andexperiment. The energies are given in atomic units.

Atom

HHeLi+

'Reference 17.

—Eo

0.47272.84777.2227

—El

0.50642.88057.2552

—El

2.891 94 2.823 79

—E (exact)'

0.527 752.903 727.279 91

sevenfold integral. This integral was numerically evalu-ated by first subdividing the integral volume into eightsubvolumes beginning with a core subvolume. The nu-merical evaluation of W& and T, integrals was performedon a Cray X-MP computer system. The results of theevaluation of A, , for the case of helium are shown in TableII. We see k, is minimized when a&

——1.63. With thisvalue of a, , A, , is recomputed with a value of E such thatA,

&——1/H. We then find E, = —2. 891 94 a.u.In Table III, we compare the computed and exact

values of the energies of three heliumlike atomic systemscorresponding to the atomic numbers Z =1,2, 3. The ex-act energies have been given previously, ' while thevalues of Eo are based on the formula of McWeeny andCoulson. Values for E,&2 and E, are based on the FIWFof (4.8).

VI. COMPTON PROFILE, KINETIC ENERGY,AND VIRIAL THEOREM

An important check on the applicability and correct-ness of the momentum-space helium-atom first-iterationwave function is a comparison of the computed Comptonprofile Jz(q) with the experimentally determined profile.

Xd Hilzzdpz (6.1)

where N is the normalization

N =8m f f f ~

P'(pi, pz) ~pidpisin8id8gzdpz,

(6.2)

and P'(p~, pz) is the FIWF with a =1.63 andE = —2.89194 a.u. We compare the calculated and themeasured' helium-atom Compton profile in columns 2and 6 of Table IV. We can see in the comparison that the

The Compton profile for a single electron in a two-electron atomic system is defined as

Ji(q)= —,' f dp,1(p)

q Pwhere I(p) is the probability that either electron hasmomentum p regardless of its direction or the value ofthe momentum of the other electron. ' We can write theCompton profile for the two-electron system asJz(q) =2J, (q), and if we align the z axis of the coordinatesystem along pz, then the angle between p, and pz is 8, sothat

Jz(q)=(gm IN) f "f"f~P'(p&, pz)

~ p, dp, sin8,q 0 0

TABLE IV. Comparison between theoretical and experimental Compton profiles for He.

q(a.u. )

00.1

0.20.30.40.50.60.70.80.91.01.21.41.61.82.02.53.0

SWFa =1.6875

1.0060.9950.9650.9160.8540.7820.7040.6250.5480.4750.4080.2950.2090.1470.1020.0720.0300.014

FIWFa =1.63

1.0381.0260.9910.9380.8690.7900.7060.6220.5400.4650.3960.2830.1990.1390.0970.0690.0300.014

HeSWF

a =1.63

1.0411.0290.9950.9420.8740.7950.7110.6270.5450.4690.3990.2840.1980.1380.0950.0660.0280.012

SWFa =1.615

1.0511.0391.0040.9490.8790.7990.7130.6270.5440.4670.3970.2810.1960.1350.0930.0650.0270.012

SWFa =1.604

1.0581.0461.0100.9550.8830.8010.7150.6270.5440.4660.3950.2780.1940.1330.0920.0630.0260.012

Experiment'gas

1.0581.0471.0090.9560.8790.8030.7100.6200.5320.4490.3810.2720.1900.1270.0920.0590.0260.007

'Reference 19.


calculated profile has a relative error of —1.89% at q =0and 100% at q =3.

The average value of the kinetic energy ( T ) of theatomic electrons in heliumlike atomic systems can be cal-culated using the FIWF. In atomic units

X dp &sinL9&d 0@2dp2,

where P'(p„pz) is the FIWF and N is given in (6.2). InTable III, we compare the energies E„(H), and E ex-perimental for a helium atomic system. The FIWF wasused to evaluate E, and (T), while (H) was obtainedfrom ( T ) by use of the virial theorem.

VII. DISCUSSION AND CONCLUSIONS

In this concluding section we would like to place ourwork in perspective with the previous work of McWeenyand Coulson as well as to discuss our results and state theconclusions we have drawn. In this paper, we set out todevelop a wave function for heliumlike atomic systems inmomentum space in order to be able, eventually, to mod-el LiH in momentum space as well. This became our ob-jective because of the following prior work. Hurst haddeveloped an excellent coordinate-space wave functionfor LiH. The momentum-space form of this wave func-tion was then used by others to calculate the LiH Comp-ton profile. The theoretical and experimental Comptonprofiles were compared, however, and found not to agree.We concluded that excellent results in coordinate spacedo not necessarily always carry over into momentumspace, and some effort might have to be made to considerthe helium problem in momentum space.

We surveyed the work already performed to develop awave function for heliumlike atomic systems in momen-tum space and found the work of McWeeny and Coulsonto be an excellent starting point because they had accom-plished much in their effort. They approached the devel-opment of a wave function for heliumlike atomic systemsin momentum space by using the variation-iteration tech-nique of Svartholrn. This technique provides a method toobtain a first-iteration wave function as well as a second-iteration wave function (SIWF). It was our hope to ob-tain a FIWF which would be accurate enough to avoidthe necessity of calculating a SIWF. With the help ofFeynman's integral evaluation technique, we were able toobtain a FIWF.

We would now like to discuss the results associatedwith this function as well as the results that might be ex-pected from the SIWF. The last step in the evaluation ofthe E& energy is to determine the value of E in A.

&so that

A, , becomes equal to 1/n. . An initial value of E is chosenin A, , to start this process on the Cray supercomputer.We chose this initial value of E in A, , based upon an inter-polation of points on the ko, k&/2, and A, , curves as well asthe corresponding differences of the energies associatedwith these curves. The points on the A. curves are chosenat the same value of a near the minimum value of A, . Ifd, and dz are the differences in values between corre-

sponding points on the A,o and A, , /2 curves and the A,o andcurves, respectively, we have d, l(Ep E—, zz )

=d2 l(Ep E~ ). This approach proved to be an excel-lent method of selecting the initial value of E in A, , todetermine E, because it was very near in value to the E,actually found. For helium the difference between E ini-tial and E, was &0.04%. This brings us to the hydrideE, energy.

Because of limited availability of the Cray X-MPsupercomputer, we did not have the opportunity to com-plete the last step in the evaluation of the E& energy forthe H hydride ion on the Cray. We did, however, esti-mate the E& energy for the H hydride ion by interpola-tion of the A,o, A, &/z, and A, , curves for this case. We ob-tained E& ———0.51222 a.u. for the hydride ion. We now

go on and estimate the E2 energy for helium.The helium E, /2 energy represents 58.5% of the

difference between the starting energy and the experimen-tal energy. The helium E& energy represents 78.9% ofthe difference between the starting energy and the experi-mental energy, or E, is one-half of the difference betweenthe E, /2 energy and the experimental energy. We wouldexpect that at best the helium E3/2 energy would be nomore than one-half of the difference between the helium

E, energy and the experimental energy because of theconditio~ that ~0 & ~1/2 & ~1& ~3/2 & ~2 & ' k. Wewould also expect that at best the helium E2 energywould be no more than one-half of the difference betweenthe E3/2 energy and the experimental energy. Conse-quently, we arrive at a lower bound to the helium Ez en-

ergy. We obtain E2 & —2.90078 a.u. ; this lower boundto the helium E2 energy is the minimum energy we couldexpect from the SIWF. We will also need an estimate ofthe value of the SIWF variational parameter a2 in orderto estimate its Cornpton profile.

In a manner analogous to the calculation of the E2 en-

ergy, we can estimate the possible range of values of thea2 variational parameter. We use the interpolation for-mula

(ap a])l(Ep E]):( apa2)l(Ep E2)

and obtain az ——1.619. We may also take the differencebetween ap and a, to represent 78.9% of the differencebetween ao and the minimum value of a corresponding tothe helium experimental energy. We then obtaina2 ——1.615. We conclude 1.619& a2 & 1.615. We will usethis range of values of a2 to estimate the SIWF Comptonprofile.

We may conclude the following regarding the calculat-ed energies. The E, energies for the H hydride ion andthe helium atom are substantial improvements over theirrespective starting energies Eo but are not close enoughin value to their respective experimental energies to beprecisely accurate. The lower bound estimate to the E2energy for helium may not be close enough in value tothe experimental energy for E2 to be precisely accurateeither.

It is well known in coordinate space that the value ofthe scale parameter or the variational constant can be ad-


justed so that the resulting average energy obtained fromuse of the virial theorem agrees with the experimental en-

ergy. We see a somewhat similar property for the start-ing wave function (SWF) and also the FIWF. For in-

stance, the average kinetic energy of the helium atom inmomentum space using the SWF is (T)=a . Conse-quently, we can use the virial theorem and seta =1.70403 to obtain (H ) = —2. 90372 a.u. A similarproperty holds for the FIWF as well.

In the same manner we can also adjust the value of thescale parameter or variational constant in the SWF to ob-tain a Compton profile which agrees well with the experi-mental profile. We see this property reAected in columns5 and 6 of Table IV. We list in column 5 the Comptonprofile using the SWF with a =1.604, and in column 6we see the experimental Compton profile of a helium gasas determined by Eisenberger. '

We may also adjust the scale factor in the Comptonprofile of this SWF and obtain the Compton profile of theFIWF. This property is shown in columns 2 and 3 inTable IV. We list in column 2 the Compton profile usingthe FIWF and list in column 3 the Compton profile usingthe SWF with a =1.63. Based on this property, we mayuse the SWF to estimate the Compton profile that may beassociated with the SIWF. We show this in column 4 ofTable IV. We list in column 4 the estimated Comptonprofile to be associated with the SIWF by using the SWF.We also compare in Table IV the Compton profile of theSWF and the FIWF. We show this in columns 1 and 2,respectively, of Table IV. Both of these profiles may becompared to the experimental profile in column 6.

We may conclude the following regarding the Comp-ton profile. The helium Compton profile of the FIWF isa substantial improvement over the helium Comptonprofile of the SWF but is not precisely accurate in com-parison to the experimental profile. The estimatedCompton profile of the SIWF is not close enough in valueto the experimental profile to be precisely accurate either.

At this point we would like to state our conclusions.By comparison of our results with experiment, we con-clude our FIWF results are not precisely accurate; how-ever, when we compare our results to the results of theSWF (a =1.6875), we see our results are a substantial

improvement. When we estimate the results to be associ-ated with the SIWF, we see those results as not beingclose enough to experimental results to be precisely accu-rate either. We have made, though, excellent progress to-wards the development of an accurate wave function forheliumlike atomic systems in rnomenturn space. Ourwork provides the necessary steps for the development ofa first-iteration open-configuration wave function for heli-

umlike atomic systems in momentum space. We wouldexpect that a first-iteration two-parameter open-configuration momentum-space wave function would besubstantially more accurate than the momentum-spaceform of an open-configuration starting wave function.

ACKNOWLEDGMENTS

We would like to thank the Air Force WrightAeronautical Laboratories and the Avionics Laboratory,Electronics Technology Division, for making it possibleto use the Wright-Patterson Air Force Base AeronauticalSystems Division Cray X-MP supercomputer.

APPENDIX: EVALUATIONOF THE ELECTRON-ELECTRON

INTERACTION INTEGRALS

2 2x&

——p&+a2 2x2 =p2+a2 2 2 2x3 po+p1+p2& po = —2E,

x4 (Pl+P2) Pl +5 2+ Pl P22 2 2

2 2xs=P&

—P22 2x6 ———x&x2x4+a x5,

x7 ——x4+4a x4,2 2

I 2x8 =x6/x2 =a]

x9 x —2a xs /X2 bl2

(Al)

x)o=Q —x,

[ sin-'x, /QX7

+sin '(X4+a X3x5/x lx2)/Qx7]

x1 1 0/x7+ Qx1 /x2x7+2X9/ax7

x&2 ——4x& /3ax2x7+2x9/3a x7 —Sa x&& /3x73 2

x 14 ——I /3a —x 9x12 /2a5 2

With the use of (Al) we can write L6 of (3.27) as

In this appendix, we sketch the evaluation of integralL6 of (3.27). Evaluation of L7 and Ls in (3.28) and (3.29)can be performed in the same manner as integral I.6. Theevaluation of these integrals can be made less cumber-some if we define a set of quantities shown below. Thesedefinitions make it easier to write the wave function andperform numerical evaluations. We have the following:

3 1 (u —u )duo [(x5/x2)u+1][x4( —u +u)+a ]" '+

In (A2) we let u = 1/y then L6 can be written

3 1 (y' —y')dy[(x5/x2+y)][x4( —1+y)+a y ]'' '+

(A2)


If we let x =(x~/x2)+y in (A3), we obtain

3 1 X dx OO X dXL6 ———

8 X 1+x5/x2 g (1/2)+2—(1+3x5/x2)

5 2 1+x5/x2 R (1/2)+2 +5 2

2X5 00 dx

2 (1+x'/x2) + 5/ 2 XR""'+'X2 5 2X

X2(3x, /x2+2) f +x5/ ~R,

(A4)

where

R1=Q1+b1X +C1X2

with

a, =xs,

(AS)

In order to evaluate the integrals in L6 of (A4), wemust examine the coefficients of R, of (AS) and its corre-sponding discriminant. If we use (Al), we see a, =xs andis a function of x4 and x 5. The discriminant51——41a1c1—b1 ———x7 depends on x4. If X4&0 then2=

61 X9

a, =xs &0

51=X7 (0 (A7)

and

2C1=Q (A6)With the results of (A7) we may now evaluate L6 of

(A4) using standard integrals. We obtain

3 1L =—8 X2

X1 X9X14 X8X12+2a'x, 4a' 2a'

X5 1(I +x&/x2)X 3Q xs

X5—(1+3x5/x2)x~4+ (3x5/x2+2)x~zX2

X9X12 2ax 5X9 4a 2x 9 X 1p2

2 + 2 2+2xs x2x7x 8 x7xs ax7x 8 x 8

(A8)

In L6 we restrict the arcsine functions (found in the quantity x,o) to range in value from —m. /2 to m/2 in order that thewave function be single valued. Evaluation of integrals L7 and Ls of (3.28} and (3.29) may be performed in the samemanner as evaluation of L&. If we assume x4+0, L7 and Ls become, respectively,

1 1L7 ——— x„—(1+2x5/x2)

X

X1 X5 x2 2ax9+ + +

x2 x2 axlx8 x7x8

2axsx9 4a 2x 9 X 1p2

+ +X2X7X8 X7 QX7X9 XS

a(3x9 —3a xs) x5 3x9xlo2 2

X7X8 X2 2xs2 2 (A9)

and

1Ls=3

X2

ax 5

X1Xs

X9X1p

2xs

X1

x42

QX2 3QX92

2 + 22x,xs 4x s

3X9 Q22

+8x 8 2xs

3QX2X9

4X1X82

X1p e (A10)

We have now evaluated L6, L7, and Ls of (3.27) —(3.29)using standard integrals, four of which are shown in (A4).All of the standard integrals used in the evaluation of L6,L7, and Ls have been evaluated on the basis that a, and

b~ are not zero in R, of (AS).It is important to understand that the integrals of

(3.27)—(3.29) have integrands which depend on themomentum of each electron. Consequently, the quanti-ties x4 and x& which are not variables of integration in

L6, L7, and Ls change value as the momentum of eachelectron changes. As p, ~—p2 then X4~0. When thishappens X5~0 also. Consequently, the very form of theintegrands of L6, L7, and Ls changes also. We have nu-merically evaluated the expressions (A8), (A9), and (A10)as a function of x4 and have found x4 can take on thevalue roughly as small as 0.001 (in a.u. ); then these in-tegrals still remain valid. But when x4 takes on valuesroughly less than 0.001 (in a.u. ) the values of x4 and x~might as well be taken as 0 in (3.27} in which case L6 be-comes


L6=, f (u —u )du,8x2a

(A11)1 1

L =—2 6x a

(A13)

with similar forms for L7 and L8. In this case L6, L7,and L8 become and

3 1L8 6x,a' ' (A12)

1L8=6x 2a

3 (A14)

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helium momentum-space wave function and compton profile

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