Using the Screened Coulomb Potential To Illustrate the Variational

Article pubs.acs.org/jchemeduc

Using the Screened Coulomb Potential To Illustrate the Variational Method José Zúñiga,* Adolfo Bastida, and Alberto Requena Departamento de Química Física, Universidad de Murcia, 30100 Murcia, Spain S Supporting Information *

ABSTRACT: The screened Coulomb potential, or Yukawa potential, is used to illustrate the application of the single and linear variational methods. The trial variational functions are expressed in terms of Slater-type functions, for which the integrals needed to carry out the variational calculations are easily evaluated in closed form. The variational energies of the first two states of the screened Coulomb potential are calculated by minimizing them with respect to the exponential parameters included in the Slater functions using elementary minimization algorithms. The project allows students to concentrate on the fundamentals and the performance of the variational method at its increasingly accurate levels of application to realistic systems.

KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Problem Solving/Decision Making, Computer-Based Learning, Quantum Chemistry, Theoretical Chemistry, Atomic Properties/Structure, Computational Chemistry, Undergraduate Research

S

electronic structure calculations, such as Hartree−Fock, configuration interactions and so forth, by performing atomic calculations and has received lesser pedagogical coverage.33,36 To describe in a unified way how the variational method works at each increasingly accurate level of application, it is convenient to have a realistic model that meets all the possibilities of the method. In our teaching practice with both undergraduate and graduate students, we have found that the screened Coulomb potential with Slater-type orbitals (STO) used as trial variational functions is certainly a good model for this purpose. From the practical point of view, this model allows the students to solve all the integrals needed to carry out the variational calculations very easily in closed form and to optimize the orbital exponents of the Slater functions acting as nonlinear variational parameters with basic minimization algorithms. From the conceptual point of view, students may then concentrate essentially on the basic ideas underlying the variational method and on the performance of it at its different stages of application. The use of the screened Coulomb potential moreover enables the students to visualize some of the new phenomena emerging when passing from the hydrogen atom to many-electron atoms, such as electronic shielding and the lifting of the degeneracy associated with the angular quantum number of the electron. The project presented in this paper additionally admits a number of extensions that may reinforce and complement the practice of the variational method.

ooner or later chemistry students learn in quantum chemistry or related courses that the elucidation of the structural and dynamic properties of matter at the microscopic, or we should say nanoscopic, scale does necessarily suppose solving the Schrödinger equation, and that, except for a few simple but extremely important models such as the particle in a box, the harmonic oscillator, the rigid rotator, and, indeed, the hydrogen atom, the solutions of the Schrödinger equation have to be found by using approximation methods. The variational method, along with perturbation theory and numerical integration of the Schrödinger equation, is one of the most powerful tools employed in this context, and its study is accordingly mandatory in quantum chemistry courses, as is its presentation in the textbooks covering the subject.1−6 This is also why the variational method has received considerable attention in this Journal.7−36 A majority of the pedagogical applications published on the variational method have concentrated fundamentally, on the one hand, on illustrating the variational principle using a single trial function that includes one or more nonlinear parameters to be optimized,7−18 and, on the other hand, on developing the linear variational method in its most common form in which only the coefficients of the linear combination which shapes the variational function are optimized.19−32 There is, however, a more elaborate version of the variational method that consist of writing the trial variational function as a linear combination of well-behaved functions, the so-called basis functions, and including in them a certain number of nonlinear parameters that are also variationally optimized. This method is employed to establish the standard basis functions routinely employed in © XXXX American Chemical Society and Division of Chemical Education, Inc.

A

dx.doi.org/10.1021/ed2003675 | J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

■

Article

SINGLE VARIATIONAL TREATMENT The screened Coulomb potential for hydrogen-like atoms is defined as follows

d W (ζ ) =0 dζ

−αr / a 0

2

V (r ) = −

requires the application of the minimum condition for W(ζ) given by

Ze e 4πε0 r

which using eq 7 yields

(1)

d W (ζ ) = dζ

where r is the distance of the electron to the nucleus, Z is the atomic number, e is the proton charge, ε0 is the vacuum permittivity, a0 = 4πε0ℏ2/(mee2) is the Bohr radius, and α is the screening parameter. This potential was introduced by Yukawa37 to account for nuclear interactions and has become widely used in other areas such as plasma38 and solid-state39 physics, and, certainly, atomic and molecular structure.40 By assuming that the nuclear mass is infinitely greater than the electron mass, the radial Schrödinger equation1 for the screened Coulomb potential, in atomic units (ℏ = me = e = a0 = 1), is written in the form

ζ[8ζ 3 + 4(3α − 2Z)ζ 2 + 6α(α − 2Z)ζ + α 3] (2ζ + α)3

Because ζ/(2ζ + α) ≠ 0, this condition is satisfied when the factor in brackets in the numerator equals zero, that is, when 3

8ζ 3 + 4(3α − 2Z)ζ 2 + 6α(α − 2Z)ζ + α = 0

(10)

Students realize that this is a cubic equation to be solved in the variational parameter ζ and they are encouraged to find its roots by using some Internet utility providing cubic equation roots. They are then expected to judiciously select the positive root which gives the minimum of W(ζ) and, by substituting it into eq 7, to determine the corresponding variational energy which, according to the variational principle, is the closest upper limit to the exact energy for this type of trial function. The variational calculations are carried out for the screened Coulomb potential with Z = 1 and screening parameters α = 0, 0.2, 0.5, 0.8, and 1.0, which potential profiles are shown in Figure 1. Students are asked first to calculate both the

(2)

where Enl and Rnl(r) are the energy levels and the radial wave functions of the potential, and n and l are the principal and angular quantum numbers. We focus our attention on the s states of the screened Coulomb potential, that is, on the states with l = 0, for which the radial Schrödinger equation (eq 2) reduces to ⎡ 1 ⎛ d2 2 d⎞ Ze−αr ⎤ ⎥R n0(r ) = En0R n0(r ) ⎢− ⎜ 2 + ⎟− r dr ⎠ r ⎦ ⎣ 2 ⎝ dr

(9)

=0

⎡ 1 ⎛ d2 l(l + 1) 2 d⎞ Ze−αr ⎤ ⎥R nl(r ) ⎢− ⎜ 2 + − ⎟+ 2 r dr ⎠ r ⎦ 2r ⎣ 2 ⎝ dr = EnlR nl(r )

(8)

(3)

In the first part of the project, students are required to find the variational energy of the 1s ground state by employing as a trial variational function the normalized 1s wave function of the hydrogen-like atom (α = 0) with the atomic number Z replaced by the variational parameter ζ, that is, ϕ1s(r ) = 2ζ 3/2e−ζr

(4) 1

The variational integral is then given by W=

∫0

∞

ϕ*1s (r )Ĥ ϕ1s(r )r 2 dr

1 2

∫0

−Z

∫0

=−

∞

⎛ d2 2 d⎞ 2 ϕ*1s (r )⎜ 2 + ⎟ ϕ (r )r d r r dr ⎠ 1s ⎝ dr

∞

ϕ*1s (r )

e −αr ϕ (r )r 2 d r r 1s

Figure 1. Screened Coulomb potential with Z = 1 for different values of the screening parameter α.

(5)

and the specific integrals that appear in this expression are easily evaluated in closed form using the result

∫0

∞

r ne−qr dr =

n! qn + 1

nonoptimized variational energies obtained using ζ = Z = 1 and the optimized variational energies obtained by minimizing W with respect to ζ and to compare their results with the accurate energies that are taken from ref 41. They organize their data in a table, as in Table 1, where they also include the differences between the variational and the accurate energies. Thus the students quickly realize the great improvements achieved in the variational energies upon minimization with respect to the variational parameter ζ. They are also asked to make plots of the variational energy W(ζ) versus ζ such as the one shown in Figure 2A for α = 0.5 for which the minimum is located at ζ = 0.87349.

n > −1; q > 0 (6)

which provides the following expression for the variational energy, W (ζ ) =

ζ2 ⎡ 8Zζ ⎤ ⎥ ⎢1 − 2⎣ (2ζ + α)2 ⎦

(7)

The optimal value of variational parameter ζ is determined by minimizing the variational energy W(ζ) with respect to ζ. This B



Article

Table 1. Variational Energies for the 1s Ground State of the Screened Coulomb Potential with Z = 1 and Different Values of the Screening Parameter α, As Obtained Using a Hydrogen 1s Variational Function

a

α

Evar./hartreea

0.0 0.2 0.5 0.8 1.0

−0.50000 (0.00000) −0.32645 (0.00036) −0.14000 (0.00812) −0.01020 (0.03450) 0.05556 (0.06585)

d

Evar./hartreeb

Eacc/hartreec

ζ

−0.50000 (0.00000) −0.32673 (0.00008) −0.14651 (0.00161) −0.03847 (0.00623) 0.00000 (0.01029)

−0.50000 −0.32681 −0.14812 −0.04470 −0.01029

1.00000 0.97568 0.87349 0.69903 0.50000

Nonoptimized variational energies (ζ = 1). bOptimized variational energies (ζ values given in the last column). cAccurate values from ref 41. Differences between variational and accurate energies are given in parentheses.

d

Inspection of Table 1 reveals that the optimal value of ζ decreases as the screening parameter α rises, which is in agreement with the usual interpretation of exponential parameter ζ as the effective nuclear charge that the screened electron feels, and also that the difference between the variational and the exact energy increases as the screening parameter α does. Students then verify that the larger the screening, the more difficult it becomes for the single variational function employed (eq 4) to account for the exact wave function. They thus find that for α = 1 the minimum of the variational energy, located at ζ = 0.5, takes the value W = 0, that is, it reaches the continuum threshold, and that for values of α higher than 1, the variational energy W(ζ) no longer has a minimum, as shown in Figure 2B for α = 1.1. This indicates that the single variational function yields acceptable results for the ground state of the screened Coulomb potential solely up to α = 1. This first part of the project is one of the required tasks that chemistry undergraduate students at our institution carry out in the quantum chemistry course. The course is given during the first semester of the third year and is typically taken by some 30 students. This particular project is done after teaching the approximation methods about midway through the semester. Students work on the project at home and are free to help each other, although the reports are submitted individually and fall due in a period of about 10 days. The reports are reviewed and graded, and returned to the students in a seminar class in which both the results and conclusions obtained and the scientific presentation of the project are discussed. Because students can help each other during the execution of the project, they usually succeed in getting the correct numerical results but struggle when it comes to extracting the proper conclusions and writing up the report in a clear, orderly way. These presentation problems also show up in the rest of tasks carried out by the students through the course, and they are corrected to a great measure in the successive elaboration of the different tasks, without the need for the students to resubmit the revised reports.

Figure 2. Variational energy W(ζ) for the 1s ground state of the screened Coulomb potential, calculated using a single STO trial function, versus the nonlinear exponential parameter ζ.

where ϕ1s is the 1s hydrogen-like function previously used, which is rewritten now as follows

■

ϕ1s(r ) = 2ζ1s3/2e−ζ1sr

LINEAR VARIATIONAL TREATMENT The systematic way to improve the variational energy provided by a single variational function consists of using the linear variational method, in which the variational function is written as a linear combination of a number of linearly independent and acceptable functions, the so-called basis functions, and the coefficients of the linear combination are optimized variationally.1 We consider then in this second stage of the project the use of the following linear combination, ϕ = c1ϕ1s + c 2ϕ2s

and ϕ2s is the normalized function 2 5/2 −ζ2sr ϕ2s(r ) = ζ2s re 3

(12)

(13)

These functions are the first two members of the so-called Slater-type orbital (STO) basis set.1 Students’ attention is called now to the fact that besides the linear coefficients c1 and c2, the variational wave function ϕ depends also on the two nonlinear parameters ζ1s and ζ2s, which can be additionally optimized.

(11) C



Article

The minimization of the variational energy with respect to the linear coefficients c1 and c2 leads to the following system of linear homogeneous equations in the unknowns c1 and c2,1 (H11 − S11W )c1 + (H12 − S12W )c 2 = 0 (H21 − S21W )c1 + (H22 − S22W )c 2 = 0

S12 = 8 3

(14) (15)

where Sij and Hij are the overlap and Hamiltonian integrals, respectively. This system has a solution different from the trivial one, c1 = c2 = 0, when the determinant of the coefficients vanish, that is, when H11 − S11W H12 − S12W H21 − S21W H22 − S22W

H11 − W

H12 − S12W

H12 − S12W H22 − W

H22 =

ζ2s2 ⎡ 1 16Zζ2s3 ⎤ ⎢ − ⎥ 2 ⎣3 (2ζ2s + α)4 ⎦

(20)

⎤ 4 3/2 5/2⎡ ζ1s(ζ1s − 2ζ2s) 2Z ⎥ ζ1s ζ2s ⎢ + 4 3 3 (ζ1s + ζ2s + α) ⎦ ⎣ (ζ1s + ζ2s)

By expanding now the secular determinant (eq 17), we get the following quadratic equation in the unknown W, 2 2 (1 − S12 )W 2 + (2H12S12 − H11 − H22)W + H11H22 − H12

=0 (17)

(22)

=0

whose roots are given by

2 2 1/2 H11 + H22 − 2H12S12 ± [(2H12S12 − H11 − H22)2 − 4(1 − S12 )(H11H22 − H12 )] 2 2[1 − S12 ]

W − ≡ W1(ζ1s , ζ2s) ≥ E1s

(24)

W+ ≡ W2(ζ1s , ζ2s) ≥ E2s

(25)

(23)

−0.5 hartree. In fact, by setting ζ1s = 1 in W1 ≡ W− as given by eq 23, students verify that W1 effectively provides the exact energy, irrespective of the value of the other exponential parameter ζ2s (see the Supporting Information). Minimization of the variational energy W1(ζ1s,ζ2s) with respect to the nonlinear parameters, ζ1s and ζ2s gives the results included in Table 2. By comparing the variational energies thus

where W− < W+. These variational energies are upper bounds to the exact energies of the ground 1s state and the first-excited 2s state, respectively, for whatever values of the nonlinear parameters ζ1s and ζ2s, that is, we have6,42

Table 2. Minimization of the Variational Energy of the 1s Ground State of the Screened Coulomb Potential with Z = 1 for Different Values of the Screening Parameter α, Using a Linear Combination of Two STO 1s and 2s Functions

where E1s and E2s are the exact energies. Minimization of each variational energy, W1 and W2, separately with respect to the nonlinear parameters ζ1s and ζ2s provides then the best variational estimates for the exact energies resulting from the linear variational function given by eq 11. The search for the variational energy minima is carried out now by using the basic gradient descent method.43 It is explained to students how this method works and they are required to write a code to find the energy minima. In our running of the project for postgraduate students, the programs are written in Fortran 77 because postgraduate courses at our institution are research oriented. For undergraduate courses, however, high-level programming packages are preferred such as Mathematica, Maple, or MathCAD, which are better suited to computational exercises in the chemistry curriculum. When considering the minimization of the two-parameters variational energies W(ζ1s,ζ2s) of the screened Coulomb potential, it is pedagogically convenient to analyze first the α = 0 case corresponding to the hydrogen atom, for which the exact energies and wave functions are known. For the 1s state, the exact hydrogen radial wave function is,1 R1s = 2e−r

(19)

(21)

and the overlap and Hamiltonian integrals are easily evaluated in closed form again using eq 6 to give the results W± =

(18)

ζ1s2 ⎡ 8Zζ1s ⎤ ⎥ ⎢1 − 2⎣ (2ζ1s + α)2 ⎦

H12 = −

Because ϕ1s and ϕ2s are both real and normalized, we have S12 = S21 and S11 = S22 = 1, and because the Hamiltonian operator is Hermitian, we additionally have H21 = H12. The secular equation (eq 16) thus becomes

(ζ1s + ζ2s)4

H11 =

=0 (16)

ζ1s3/2ζ2s5/2

a

α

ζ1s

ζ2s

Evar(ζ1s,ζ1s) /hartree

Eacc/ hartreea

(Evar − Eacc) /hartree

0.0 0.2 0.5 0.8 1.0 1.1

1.00000 1.08366 1.11783 1.04208 0.90890 0.77961

0.94932 0.80679 0.59124 0.40526 0.28262

−0.50000 −0.32681 −0.14811 −0.04457 −0.00975 −0.00133

−0.50000 −0.32681 −0.14812 −0.04470 −0.01029 −0.00229

0.00000 0.00000 0.00001 0.00013 0.00054 0.00096

Accurate values from ref 41.

obtained with those given in Table 1, students clearly notice the large improvements achieved upon inclusion of the new basis function ϕ2s(r) in the variational calculations. Even better, the linear variational function (eq 11) now provides acceptable energies for the 1s levels with α ≥ 1, as shown in the α = 1.0 and α = 1.1 cases included in Table 2 for which the single variational wave function fails. We should also note that when searching for the variational minima, students find an additional local minimum for every value of α, the nature of which is examined and discussed in the Supporting Information.

(26)

and the variational energy W1(ζ1s,ζ2s) is therefore expected to have a minimum at ζ1s = 1 providing the exact energy E1s = D



Article

As far as the minimization of the variational energy W2(ζ1s,ζ2s) for the excited 2s state is concerned, the exact hydrogen radial wave function (α = 0) in this case is1 1 ⎛⎜ r⎞ 1 − ⎟e−r /2 R 2s = ⎝ 2 2⎠

(27)

a

ζ2s

Evar(ζ1s,ζ1s) /hartree

Eacc/ hartreea

(Evar − Eacc) /hartree

0.00 0.05 0.10 0.15 0.20 0.25

0.50000 0.50046 0.75815 0.83515 0.90375 0.97165

0.50000 0.49155 0.42665 0.37392 0.31450 0.24600

−0.12500 −0.08175 −0.04992 −0.02718 −0.01196 −0.00297

−0.12500 −0.08177 −0.04993 −0.02722 −0.01211 −0.00340

0.00000 0.00002 0.00001 0.00004 0.00015 0.00042

(30)

c2 = ∓

a [a + b − 2abS12]2

(31)

2

α

c1s

c2s

α

c1s

c2s

1.00000 0.86409 0.73029 0.62380 0.52850 0.44725

0.00000 0.16325 0.34266 0.50798 0.65516 0.76337

0.00 0.05 0.1 0.15 0.2 0.25

1.00000 0.96168 0.49042 0.38211 0.20049 0.19625

−1.73205 −1.69489 −1.18323 −1.09982 −1.04787 −1.01676

Having the radial wave functions available, we can calculate the average values of any physical quantity of interest. It is interesting, for example, to consider the average value of the radial distance r that, according to eq 11, is given by ⟨r ⟩ =

As noticed, the project is designed to optimize separately the energies of the 1s and 2s levels with respect to the orbital exponents ζ1s and ζ2s. For a given value of the screening parameter α, this strategy produces nonorthogonal wave functions, because each of them depends on a different pair of orbital exponents (ζ1s,ζ2s). This is clearly observed in Tables 2 and 3 for α = 0.2. It is also possible to obtain orthogonal wave functions for the 1s and 2s states by minimizing, for example, the sum of the variational energies, W1 + W2, with respect to the orbital exponents ζ1s and ζ2s. The variational energies obtained then for α = 0.2 are W1 = −0.32675 hartree and W2 = −0.01189 hartree, with ζ1s = 0.97570 and ζ2s = 0.30658, and they are somewhat higher, and therefore less accurate, than those given in Tables 2 and 3 calculated by separate minimization of the variational energies. Once the variational energies W1 and W2 have been determined, solution of the linear homogeneous eqs 14 and 15 for each variational energy provides the corresponding c1 and c2 coefficients of the wave functions. These two equations are equivalent due to their homogeneous character, so only one of them, let us say the first, can be employed. The other equation needed is taken from the normalization condition of the variational wave function ϕ*(r )ϕ(r )r 2 dr = 1

∫0

∞

ϕ*(r )rϕ(r )r 2 dr

∫0

+ |c 2|2

∞

ϕ*1s (r )rϕ1s(r )r 2 dr

∫0

+ 2c1c 2

∞

∫0

ϕ*2s (r )rϕ2s(r )r 2 dr +

∞

ϕ*1s (r )rϕ2s(r )r 2 dr

(32)

where the integrals appearing here are again easily evaluated using eq 6 to yield ∞ 3 ϕ*1s (r )rϕ1s(r )r 2 dr = 2ζ1s 0 (33)

∫

∫0

∫0

∞

∞

ϕ*2s (r )rϕ2s(r )r 2 dr =

5 2ζ2s

(34)

ϕ*1s (r )rϕ2s(r )r 2 dr = 32 3

ζ1s3/2ζ2s5/2 (ζ1s + ζ2s)5

(35)

The average radial distances calculated for the 1s and 2s states of the screened Coulomb potential are given in Table 5. Table 5. Average Radial Distances for the 1s and 2s States of the Screened Coulomb Potential with Z = 1 for Different Values of the Screening Parameter α

(28)

which, using the expansion 11, yields c12 + c 22 + 2c1c 2S12 = 1

2s State

0.0 0.2 0.5 0.8 1.0 1.1

= |c1|2

∞

2

1s State

Accurate values from ref 41.

∫0

2

Table 4. Coefficientes of the Variational Wave Functions for the 1s and 2s States of the Screened Coulomb Potential with Z = 1 for Different Values of the Screening Parameter α

Table 3. Minimization of the Variational Energy of the 2s Excited State of the Screened Coulomb Potential with Z = 1 for Different Values of the Screening Parameter α, Using a Linear Combination of Two STO 1s and 2s Functions ζ1s

b [a + b − 2abS12]2 2

where a = H11 − W and b = H12 − S12W. In Table 4, we give the values obtained for the coefficients of the 1s and 2s variational wave functions.

and the linear variational function (eqs 11−13) reproduces this result for ζ1s = ζ2s = 0.5, and c1 = 1 and c2 = −√3 = −1.73205. The minimum of the variational energy W2(ζ1s,ζ2s) is therefore located at (ζ1s,ζ2s) = (0.5,0.5) and provides the exact energy value of −0.125 hartree. Variational calculations of the 2s state are carried out in this case for α = 0.05, 0.1, 0.15, 0.2, and 0.25, and the results are given in Table 3. It is observed in this table that the accuracy of the variational energies is similar to that reached for the 1s state using the linear variational function.

α

c1 = ±

(29)

where coefficients c1 and c2 are assumed to be real. On solving eqs 14 and 29 for c1 and c2, we obtain, E

α

⟨r⟩1s/a0

α

⟨r⟩2s/a0

0.0 0.2 0.5 0.8 1.0 1.1

1.50000 1.54974 1.80538 2.51532 3.99597 6.35749

0.00 0.05 0.1 0.15 0.2 0.25

6.00000 6.10193 6.51700 7.17312 8.27415 10.32791



Article

In Figure 3, we plot the radial distribution functions |ϕ (r)|2 r of the 1s and 2s states of the screened Coulomb potential

minimization gradient descent method is explained to them. Students then work in the computer lab writing, debugging, and checking the computer code needed to minimize the variational energies, assisted by the instructor. The lab work normally takes a 3−4 h evening session, although students can go back to the lab in free hours if they need more time. The reports are submitted in a period of 15 days, and then reviewed and graded valuing the scientific content, its clarity, and well argued presentation. Finally, the project is presented orally by a single student in a seminar in order to exercise and reinforce communication skills.

2

■

CONCLUSIONS AND SUGGESTIONS In this paper, we describe an application of the variational method to the screened Coulomb potential in which the socalled Slater-type functions are used to construct the trial variational wave functions. All the integrals required in this model to perform the variational calculations are easily evaluated in closed form. In addition, when two basis functions are used to write the linear variational function, minimization of the variational energies with respect to the nonlinear exponential parameters can be carried out using basic minimization algorithms. Thus, students can concentrate on the progressive applications of the variational method, from one single trial function to linear combinations containing more than one nonlinear optimization parameter, so providing increasingly accurate variational energies for realistic atomic systems. The most elementary version of the project, in which only the single trial variational function is employed, is suitable as a problem assignment for undergraduate students taking Quantum chemistry courses, which is what we do at our institution. The entire project can be also undertaken by undergraduates both individually or in groups, with guidance in the use of programming packages like Mathematica, Maple, or MathCAD to perform the variational calculations and display the results obtained properly. The second part of the project, and more elaborated versions of the project such as the one included in the Supporting Information and those suggested below, can also be used in computational or advanced quantum chemistry courses for graduate students in order to illustrate, for example, how the electronic structure calculations of atoms and molecules are made at the Hartree−Fock and higher levels. The variational treatment of the screened Coulomb potential presented in this paper can be applied in the same terms as employed here to states with l > 0 (see the Supporting Information) and admits a number of extensions and additional possibilities. First, we can use alternatively 1s-type STO basis functions with different orbital exponents to build up the linear variational function, that is, ϕ1s and ϕ1s′ basis functions, as given in eq 4 with exponents ζ1s and ζ1s′. In the framework of the electronic structure calculations, this is equivalent to using double-ζ basis set functions. Second, a number of three, instead of two, STO basis functions can also be employed to carry out the variational calculations. A third-order secular determinant is derived in this case, whose expansion provides a cubic equation in the variational energy which can be solved by similar procedures as described earlier in the paper. The algebra in this case is somewhat more elaborated but still manageable, as is the search for the minima of the variational energies. Finally, the application of the variational method to the screened Coulomb potential presented in this paper is clearly

Figure 3. (A) Radial distribution functions of the screened Coulomb potential for the 1s ground state and (B) the 2s excited state (lower panel) calculated using a linear combination of the 1s and 2s STO functions.

calculated variationally for different values of the screening parameter α. As observed in Figure 3A, the radial distribution of the 1s state spreads out as the screening parameter rises, resulting in the accelerated increase of the average radial distance observed in Table 5 as α approaches its critical value of 1.1906. Note, for example, the large delocalization of the 1s state for the largest value of α = 1.1. The average value of the radial distance in this case is 6.35759a0, and the radial density takes non-negligible values up to a distance as large as 15a0. For the 2s state, the radial distribution function (Figure 3B) exhibits two maxima, one inner and one outer. The position of the inner maximum hardly changes as α increases, only its height decreases. The outer and stronger maximum moves, however, toward higher values of r as α rises, causing the increase of the average value of the radial distance ⟨r⟩ (see Table 5) which reaches 10.32791 a0 for the highest value of α = 0.25. This second part of the project is a computational application that graduate students taking the advanced quantum chemistry course offered at our institution carry out. This course is attended by about 10 students who work individually and who have previously received some basic knowledge of Fortran programing. In a first session of approximately 1 h, they are given the outline and the objectives of the project and the F



Article

(21) Rioux, F. J. Chem. Educ. 1992, 69, A240. (22) Keeports, D. J. Chem. Educ. 1989, 66, 314. (23) Knox, K. J. Chem. Educ. 1980, 57, 626. (24) Sims, J. S.; Ewing, G. E. J. Chem. Educ. 1979, 56, 546. (25) Earl, B. J. Chem. Educ. 2008, 85, 453. (26) Dunn, S. K. J. Chem. Educ. 2002, 79, 1378. (27) Grubbs, W. T. J. Chem. Educ. 2001, 78, 1557. (28) Besalu, E.; Marti, J. J. Chem. Educ. 1998, 75, 105. (29) Summerfield, J. H.; Beltrame, G. S.; Loeser, J. G. J. Chem. Educ. 1999, 76, 1430. (30) Veguilla-Berdecia, L. A. J. Chem. Educ. 1993, 70, 928. (31) Rioux, F. J. Chem. Educ. 1982, 59, 773. (32) Harris, D. K.; Rioux, F. J. Chem. Educ. 1981, 58, 618. (33) Rioux, F. J. Chem. Educ. 1994, 71, 781. (34) Parson, R. J. Chem. Educ. 1993, 70, 115. (35) Fucaloro, A. F. J. Chem. Educ. 1986, 63, 579. (36) Snow, R. L.; Bills, J. L. J. Chem. Educ. 1975, 52, 506. (37) Yukawa, H. Proc. Phys. Math. Soc. Jpn 1935, 17, 48. (38) Margenau, H; Lewis, M. Rev. Mod. Phys. 1959, 31, 569. (39) Kittel, C. Introduction to Solid State Physics; Wiley: New York, 1986; pp 266−271. (40) Ugalde, J. M.; Sarasola, C.; López, X. Phys. Rev. A 1997, 56, 1642. (41) Stubbins, C. Phys. Rev. A 1993, 48, 220. (42) Kenny, J. E. Am. J. Phys 1994, 62, 184. (43) Jensen, F. Introduction to Computational Chemistry; Wiley: Chichester, 1999; Chapter 14. (44) Luaña, V.; de la Roza, A. O.; Blanco, M. A.; Recio, J. M. Eur. J. Phys. 2010, 31, 101.

open to the use of a higher number of basis functions, by formulating the variational problem in matrix form, in which the variational energies are obtained by diagonalizing the representation of the Hamiltonian matrix in the Slater basis set (see the Supporting Information for the Hamiltonian matrix elements), exactly in the same terms as in previously reported applications of the linear variational method to other quantum systems.19−24,35,44 The optimization of the nonlinear exponential parameters becomes in this case more troublesome, and is presumably more convenient either to limit the number of nonlinear parameters to be optimized or just to fix them, for example, at their hydrogen values, and concentrate on the analysis of the convergence of the variational energies with respect to the number of basis functions used.

■

ASSOCIATED CONTENT

S Supporting Information *

A student handout containing direct instructions for the execution of the project; discussion of the local minima for the 1s states; application to p states; general expressions for the quantum integrals of the screened Coulomb potential in the Slater basis set. This material is available via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partially supported by the Spanish Ministerio de Ciencia e Innovación under Projects CTQ2011-25872 and CONSOLIDER CSD2009-00038, and by the Fundación Séneca del Centro de Coordinación de la Investigación de la Región de Murcia under Project 08735/PI/08.

■

REFERENCES

(1) Levine, I. N. Quantum Chemistry, 5th ed; Prentice Hall: Upper Saddle River, NJ, 2000; Chapter 8. (2) McQuarrie, D. A. Quantum Chemistry; University Science Books: Sausalito, CA, 2007; Chapter 8. (3) Atkins, P.; Friedman, R. Molecular Quantum Mechanics; Oxford University Press: New York, 2005; pp 183−187. (4) Ratner, M. A.; Schatz, G. C. Introduction to Quantum Mechanics in Chemistry; Prentice-Hall: Upper Saddle River, NJ, 2001; pp 105−109. (5) Dahl, J. P. The Quantum World of Atoms and Molecules; World Scientific: Hackensack, NJ, 2009; Chapter 12. (6) Szabo, A; Ostlund, N. S. Modern Quantum Chemistry; Dover: New York, 1989. (7) Sommerfeld, T. J. Chem. Educ. 2011, 88, 1521. (8) Magnasco, V. J. Chem. Educ. 2008, 85, 1686. (9) Davis, S. L. J. Chem. Educ. 2007, 84, 711. (10) Reed, B. C. J. Chem. Educ. 1997, 74, 935. (11) Knudson, S. K. J. Chem. Educ. 1997, 74, 930−934. (12) Bendazzoli, G. L. J. Chem. Educ. 1993, 70, 912. (13) Blaise, P.; Henri-Rousseau, O. J. Chem. Educ. 1988, 65, 9. (14) Martins, L. J. A. J. Chem. Educ. 1988, 65, 861. (15) Li, W. K. J. Chem. Educ. 1988, 65, 963. (16) Reed, L. H.; Murphy, A. R. J. Chem. Educ. 1986, 63, 757. (17) Robiette, A. G. J. Chem. Educ. 1975, 52, 95. (18) Li, W. K. J. Chem. Educ. 1987, 64, 128. (19) Beddard, G. S. J. Chem. Educ. 2011, 88, 929. (20) Casaubon, J. I.; Doggett, G. J. Chem. Educ. 2000, 77, 1221. G


Using the Screened Coulomb Potential To Illustrate the Variational

Recommend Documents