American Journal of Condensed Matter Physics 2014, 4(3): 51-56 DOI: 10.5923/j.ajcmp.20140403.02

Computation of the First and Second Ionization Energies of the First Ten Elements of the Periodic Table Using a Modified Hartree-Fock Approximation Code

Abdu S. G.1, Onimisi M. Y.2,*, Musa N.3

1Department of Physics, Kaduna State University, Kaduna, Nigeria 2Department of Physics, Nigerian Defence Academy, Kaduna, Nigeria 3Department of Applied Science, Kaduna Polytechnic, Kaduna, Nigeria

Abstract The first and second ionization energies of the first ten elements of the periodic table were computed using the Hartree-Fock approximation code for small atomic systems developed by Koonin, S. E and Meredith, D. C. The results obtained compared fairly with the experimental values. The maximum and minimum values expected between helium at 31.4 eV and lithium at 3.4 eV for the first ionization energy and lithium at 86.7 eV and helium at 46.2 eV for the second ionization energy were observed. The spikes for the noble gases (helium and neon) were also observed for the first ionization energy. The computed values for the effective nuclear charge Z* had fairly, the same characteristics as the experimental results Keywords First ionization energy, Second ionization energy, Ground state energy, Hartree-Fock, Slater-determinant

1. Introduction constant divided by 2π, we get 2 2 2 = 2 + (2a) In quantum mechanics an atom can be viewed as a 2 =1 >1 many-particle system [1], [2]. While the wave function for a ℏ 𝑁𝑁 𝑁𝑁 𝑍𝑍𝑒𝑒 𝑁𝑁 𝑒𝑒 � 𝑚𝑚 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝒓𝒓�⃗𝒊𝒊 𝑖𝑖 𝒓𝒓�⃗𝑖𝑖𝑖𝑖 single particle system is a function of only the coordinates of In the calculation𝑯𝑯 − of∑ atomic∇ − structure,∑ it∑ is convenient to that particular particle and time, Ψ(r, t), a many-particle use Hartree’s atomic units (au) [4]. system will depend on the coordinates of all the particles. The unit of mass = rest mass of the electron me The Born-Oppenheimer Hamiltonian [3] for N electrons The unit of charge = the magnitude of the electronic moving about a heavy nucleus can be written as charge, e. 2 2 2 The unit of length = the first Bohr radius of the hydrogen 2 = =1 + <1 (1) atom 2� 2 𝑁𝑁 𝑷𝑷𝑖𝑖 𝑁𝑁 𝑍𝑍𝑒𝑒 𝑁𝑁 𝑒𝑒 ℏ 𝑖𝑖 𝑖𝑖 𝒊𝒊 𝑖𝑖 In these units, (2a) becomes Here, (ri) 𝑯𝑯is� the∑ location/position𝑚𝑚 − ∑ 𝒓𝒓�⃗ of∑ electrons,𝒓𝒓�⃗𝑖𝑖𝑖𝑖 m, -e, 𝑚𝑚𝑒𝑒 electron mass and charge respectively and = , 2 1 =1 + 2 + >1 = 0 (2b) the separation between electrons at i and j. 𝑍𝑍 𝒓𝒓�⃗𝑖𝑖𝑖𝑖 𝒓𝒓�⃗𝑖𝑖 − 𝒓𝒓�⃗𝒋𝒋 𝑁𝑁 𝑁𝑁 𝑁𝑁 The three sum of equation (1) embody [4] For one electron,�∑𝑖𝑖 ∇𝑖𝑖 (2b) can�∑ be𝑖𝑖 �𝒓𝒓⃗ 𝒊𝒊solved∑𝑖𝑖 exactly.𝒓𝒓�⃗𝑖𝑖𝑖𝑖 �� 𝜓𝜓 The solution a. the electron kinetic energy (the first term) corresponding to bound states is b. the electron-nucleus attraction (the second term) 1 c. the electron-electron coulomb repulsion (the third ( ) = ( ) ( ) term) Electrons are Fermions𝜓𝜓 𝒓𝒓� ℜwhich𝑛𝑛𝑛𝑛 𝑟𝑟 𝑌𝑌obey𝑙𝑙𝑙𝑙 𝑟𝑟 ̂ Pauli’s exclusion For the accuracy of the Self Consistent Field, other 𝑟𝑟 approximations were neglected such as the spin-orbit principle. This requires that the wave function of electrons interaction, hyperfine interaction, recoil motion of nucleus should be anti-symmetric with respect to the interchange of and relativity. coordinates x of any two electrons. By substituting with the quantum mechanical operator ( 1, 2, … … . , ) = ( 1, 2, … … . , ) (3) = , i, being the imaginary unit and is the Planck’s Slater determinants satisfy𝑁𝑁 this anti-symmetric 𝑁𝑁condition 𝒑𝒑� Ψ 𝑥𝑥 𝑥𝑥 𝑥𝑥 −Ψ 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝒊𝒊 through appropriate linear combination of Hartree products, *𝒑𝒑� Corresponding−𝑖𝑖ℏ𝛁𝛁 author: ℏ which are the non-interaction electron wave functions. [email protected] (Onimisi M. Y.) Published online at http://journal.sapub.org/ajcmp It may be noted here that the Slater determinantal Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved wave-function can be conveniently written as

52 Abdu S. G. et al.: Computation of the First and Second Ionization Energies of the First Ten Elements of the Periodic Table Using a Modified Hartree-Fock Approximation Code

1 = ! ( 1) ( ) ( ) ( ) (4) The total ground state energy ! 1 1 2 𝑁𝑁 𝑁𝑁 𝑝𝑝 𝑃𝑃− 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑁𝑁 | | | | | | whereΨ the√ 𝑁𝑁summation∑ − is𝑃𝑃 �over𝜒𝜒 𝑥𝑥 all𝜒𝜒 possible𝑥𝑥 ⋯ ⋯N!𝜒𝜒 nu𝑥𝑥mber� of 0 = 0 1 0 + 0 2 0 = permutations amongst the N completely identical electrons 𝑖𝑖 𝑖𝑖 and p is the parity of the permutation P. [2] 𝐸𝐸 1⟨Ψ 𝐺𝐺 Ψ ⟩ ⟨Ψ2 𝐺𝐺 Ψ ⟩ �⟨𝜓𝜓2 ℎ 𝜓𝜓 ⟩ + 𝑖𝑖 (12) 2 The many-body Hamiltonian for a system of N interacting 𝑒𝑒 𝑒𝑒 electrons in the presence of nuclei fixed in some selected ∑𝑖𝑖≠𝑗𝑗 ��𝜓𝜓𝑖𝑖𝜓𝜓𝑗𝑗 � 𝑟𝑟𝑖𝑖𝑖𝑖 �𝜓𝜓𝑖𝑖𝜓𝜓𝑗𝑗 � − �𝜓𝜓𝑖𝑖𝜓𝜓𝑗𝑗 � 𝑟𝑟𝑖𝑖𝑖𝑖 �𝜓𝜓𝑗𝑗 𝜓𝜓𝑖𝑖�� configuration can be written in the form 1 2 2. Materials and Methods = ( ) + , (5) 2 𝑁𝑁 𝑁𝑁 𝑒𝑒 The materials used for this research were: 𝑒𝑒 𝑖𝑖 𝒊𝒊 𝑖𝑖≠𝑗𝑗 𝑟𝑟𝑖𝑖𝑖𝑖 where 𝐻𝐻 ∑ ℎ 𝒓𝒓 ∑ • Fortran code developed by S. E. Koonin and D. C. 2 ( ) = + ( ) (6) Meredith in 1989 for the calculation of the ground-state 2 𝑝𝑝 energies of small atomic systems. denotes for each electronℎ 𝒓𝒓𝒊𝒊 the kinetic𝑚𝑚 𝑉𝑉 energy𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝒓𝒓 operator plus the • Origin 6.1 application software for graphical potential energy due to the nuclei Eq. (4). interpretation of the data generated from the Fortran code. 2 ( ) = • PC with Windows 7 operating system on which the In this case, | | 𝑧𝑧𝐼𝐼𝑒𝑒 application, Microsoft Developer Studio ® was mounted. where 2 is the mutual attractive potential between the 𝑉𝑉𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝒓𝒓 − ∑𝑖𝑖≠𝑗𝑗 𝒓𝒓−𝑹𝑹𝑰𝑰 It is used to run the Fortran code. electron and the nucleus and is their distance of 𝐼𝐼 The Hartree-Fock approximation code was compiled and separation.𝑧𝑧 𝑒𝑒 ® 𝒓𝒓 − 𝑹𝑹𝑰𝑰 debugged using Microsoft Developer Studio . ( , , … . . , ) 1 1 2 2 The program is interactive. The DOS window that opens = ( , , … . . , ), (7) 𝐻𝐻𝑒𝑒 Ψ 𝒓𝒓1 𝜎𝜎1 𝒓𝒓2 𝜎𝜎2 𝒓𝒓𝑁𝑁𝜎𝜎𝑁𝑁 when in operation requests for the following: where are the space and spin 𝑁𝑁variables𝑁𝑁 of the i-th • 𝐸𝐸 Ψ 𝒓𝒓 𝜎𝜎 𝒓𝒓 𝜎𝜎 𝒓𝒓 𝜎𝜎 The nuclear charge for the atomic system of interest electron. 𝑖𝑖 𝑖𝑖 • The number of electrons in the 1s, 2s and 2p states Although𝒓𝒓 𝜎𝜎 the Hamiltonian He does not include respectively spin-dependent energy terms, the electron spin plays a • The radial step size measured in Angstrom unit fundamental role since the correct many-electron • The outer radius of the lattice in Angstrom unit ( , , … . . , ) wavefunctions 1 1 2 2 must be • The number of iterations antisymmetric for interchange of the spatial and spin 𝑁𝑁 𝑁𝑁 • The choice of output for the result coordinates of any Ψtwo𝒓𝒓 electrons.𝜎𝜎 𝒓𝒓 𝜎𝜎 𝒓𝒓 𝜎𝜎 ο On the screen The many-body electron Hamiltonian (5) contains two ο Saved to a file with a filename of your choice types of operators. One is the sum of one-particle operators ο Sent to a graphical output terminal of the form The code calculates 1 = ( ) (8a) • The total Kinetic energy, Ktot; inter-electron the other type is the sum of two-𝑁𝑁particle operators of the form repulsive potential energy, Vee; Electron-nucleus 𝐺𝐺 ∑𝑖𝑖 ℎ 𝒓𝒓𝒊𝒊 1 2 attractive potential energy, Ven; Exchange potential = , (8b) 2 2 energy, Vex; Total potential energy, Vtot; Total energy, 𝑁𝑁 𝑒𝑒 Etot and the effective nuclear charge, Z*. which describes electron𝐺𝐺 -electron∑𝑖𝑖≠ 𝑗𝑗interactions.�𝒓𝒓𝒊𝒊−𝒓𝒓𝒋𝒋� The ground-state energy for the first ten elements of the Taking the expectation value of (8a) and (8b), we have: periodic table was calculated using the electronic 0| 1| 0 = | | (9a) configuration of the respective elements 2 2 𝑖𝑖 𝑖𝑖 𝑖𝑖 = 1⟨Ψ2 𝐺𝐺 Ψ1⟩ 2 ∑ ⟨ 𝜓𝜓 1ℎ 𝜓𝜓2 ⟩ 2 1 (9b) 2.1. Computing First and Second Ionization Energies 𝑒𝑒12 𝑒𝑒12 To compute the first ionization energy of an element, we and the expectation�𝜓𝜓 𝜓𝜓 � 𝑟𝑟 value�𝜓𝜓 𝜓𝜓 of� G−2 �on𝜓𝜓 𝜓𝜓a given� 𝑟𝑟 �𝜓𝜓 determinantal𝜓𝜓 � state thus becomes: remove the outer electron from the lattice by deliberately 2 2 reducing the number of electrons in the outer radius. For 1 2 0 0 0| 2| 0 = 1 2 1 2 1 2 2 1 (10a) 2 12 12 example, He has an electronic configuration of 1s 2s 2p at 𝑒𝑒 𝑒𝑒 ground-state. We now remove one electron from the 1s2 ⟨ΨWhich𝐺𝐺 Ψ can⟩ be∑ 𝑖𝑖re≠𝑗𝑗-�written𝜓𝜓 𝜓𝜓 � 𝑟𝑟 as� 𝜓𝜓 𝜓𝜓 � − �𝜓𝜓 𝜓𝜓 � 𝑟𝑟 �𝜓𝜓 𝜓𝜓 � orbital, giving us 1s1 2s0 2p0. The removal of the electrons 1 2 from the elements is shown in Table 3.2. After entering the | | = 0 2 0 2 parameter required, we then ran the code for a number of 𝑒𝑒 ⟨ ⟩ 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 iterations, all the while, looking at the result obtained. The Ψ 𝐺𝐺 Ψ 1 � �𝜓𝜓 𝜓𝜓 � 𝑖𝑖𝑖𝑖2 �𝜓𝜓 𝜓𝜓 � 𝑖𝑖≠𝑗𝑗 𝑟𝑟 (10b) iterations can be stopped immediately convergence was 2 𝑒𝑒 achieved. − ∑𝑖𝑖≠𝑗𝑗 �𝜓𝜓𝑖𝑖𝜓𝜓𝑗𝑗 � 𝑟𝑟𝑖𝑖𝑖𝑖 �𝜓𝜓𝑗𝑗 𝜓𝜓𝑖𝑖�

American Journal of Condensed Matter Physics 2014, 4(3): 51-56 53

Table 1. Electronic configuration of first ten elements of the periodic table Using the same procedure, the second ionization energy after removal of first electron was also calculated. Electronic configuration The highly simplified algorithmic flowchart of the Elements 1s occupied 2s occupied 2p occupied Hartree-Fock approximation is shown in Figure 1 below state state state Hydrogen - - - Helium 1 0 0 Lithium 2 0 0 Beryllium 2 1 0 Boron 2 2 0 Carbon 2 2 1 Nitrogen 2 2 2 Oxygen 2 2 3 Fluorine 2 2 4 Neon 2 2 5

Table 2. Electronic configuration of first ten elements of the periodic table after removal of second electron Electronic configuration Elements 1s occupied 2s occupied 2p occupied state state state Hydrogen - - - Helium - - - Lithium 1 0 0 Beryllium 2 0 0 Boron 2 1 0 Carbon 2 2 0 Nitrogen 2 2 1 Oxygen 2 2 2

Fluorine 2 2 3 Figure 1. Simplified algorithmic flowchart of the Hartree-Fock Neon 2 2 4 approximation The total energy computed for each element was recorded and the ground state energy subtracted from it. The result obtained was recorded as the first ionization energy for the 3. Results and Discussion respective element. The following results were obtained

Table 3. Computed and calculated values of Ground-State, 1st and 2nd ionization energies Experimental values Computed values [4]

2nd 1st Ionization 1st Ionization 2nd Ionization Element Ground State Ionization energy energy energy energy

Hydrogen -8.4 8.4 13.6

Helium -77.64 31.4 46.2 24.6 54.4 Lithium -183.66 3.4 86.7 5.4 75.6 Beryllium -388.38 10.9 12.6 9.3 18.2 Boron -654.16 10.8 25.9 8.3 25.2 Carbon -1006.82 13.0 16.4 11.3 24.4 Nitrogen -1455.46 15.9 23.6 14.5 29.6 Oxygen -2008.56 16.1 31.4 13.6 35.1 Fluorine -2676.80 23.2 40.0 17.4 34.9 Neon -3470.44 31.1 49.2 21.6 40.9

54 Abdu S. G. et al.: Computation of the First and Second Ionization Energies of the First Ten Elements of the Periodic Table Using a Modified Hartree-Fock Approximation Code

Table 4. Computed and Calculated Effective Nuclear charge of first ten elements of the periodic table Effective nuclear Effective nuclear Element Atomic number, Z charge Z* charge (Computed) [5] Hydrogen 1 0.849 1 Helium 2 1.844 1.6875 Lithium 3 2.937 2.9606 Beryllium 4 3.538 3.6848 Boron 5 4.39 4.6795 Carbon 6 5.135 5.6727 Nitrogen 7 5.877 6.6651 Oxygen 8 6.614 7.6579 Fluorine 9 7.351 8.6501 Neon 10 8.084 9.6421

Figure 2. Graph of Elements against ground-state energy

Figure 3. Graph of Computed and calculated 1st I.E. against Elements

American Journal of Condensed Matter Physics 2014, 4(3): 51-56 55

Figure 4. Graph of Computed and Calculated 2nd I.E. against Elements

Figure 5. Graph of Computed and Calculated Effective nuclear charge vs. Elements Table 1 gives the computed ground-state energy and the energies of the ten elements are negative and decrease with first and second ionization energies of the first ten elements. increasing atomic number. This is because the energies bind It also shows the experimental values for the first and second the electrons to the nucleus. Commensurate amount of ionization energies of the first ten elements of the periodic energy, called the binding energy will be required to remove table. the electrons bound to the respective nucleus from its From Table 1, the computed values for the ground-state position to the point where the kinetic energy is zero which

56 Abdu S. G. et al.: Computation of the First and Second Ionization Energies of the First Ten Elements of the Periodic Table Using a Modified Hartree-Fock Approximation Code implies that binding energy increases with increase in atomic 4. Conclusions number. [6] Fig. 2, gives a graphical representation of the relationship The Hartree-Fock equation as a variational method was between the ground-state energy and the elements. tested on the first ten elements of the periodic table using the From Figures 3-4, the comparison of our computed values Fortran code as developed by [4]. The characteristic zig-zig and the experimental results for the first and second graphical pattern as we traversed the ten elements was ionization energies against each element, shows a good observed. This is characteristic of all the elements as they similarity as the rise and dip at He and Li for the first have different properties that define them e.g. alkali earth ionization energy and Li and Be for the second ionization metals, noble gases, etc. energies compare favourably. From Figure 3, we can see that the first ionization energy of helium (31.4 eV) is much greater than that of Lithium (3.4 eV). This is in agreement with [7]. The electrons provide REFERENCES some shielding for each other as mutual repulsion pushes them away from the nucleus. The outer electron of lithium [1] U. Rothlisberger and I. Tavernelli, 2011, “Electronic structure occupies a new shell screened by two electrons and the methods for molecular excited states.” http://www.tddft.org/ TDDFT2008/lectures/IT3.pdf. Accessed on 3/10/2013 ionization is much less than that of hydrogen or helium. This is in agreement with [8]. [2] P. C. Deshmukh, A. Banik, and D. Angom, Hartee-Fock From Table 1, the second ionization energies have higher self-consistent field method for many-electron system. values than the first ionization energies. This is because at nptel.iitm.ac.in/courses/115106057/STiAP_Unit_4_ Hartree _Fock_L19_to_L23.pdf, 2011. Retrieved on 3/10/2013 the second ionization level, there are more protons than electrons holding the electrons down therefore more energy [3] D. D. Fitts, Principles of quantum mechanics: as applied to is required to pull out the second + = 1 22 0, 2+ = chemistry and chemical physics. Cambridge University Press, 1 1 [9]. The remarkable difference between the first and Cambrigde IRP, 2002. 𝐿𝐿𝐿𝐿 𝑠𝑠 𝑠𝑠 𝐿𝐿𝐿𝐿 second ionization energies for lithium is due to the sudden [4] S. E. Koonin and D. C. Meredith, Computational physics breaking𝑠𝑠 -in to an inner level, closer to the nucleus with less (FORTRAN version), Westview Press, 1989. shielding [10]. Table 2 shows the computed effective nuclear charge for [5] dept.astro.lsa.umich.edu/~cowley/ionen.htm Accessed on 20/2/2014. the first ten elements of the periodic table. Z* shows a steady increase from Hydrogen to Neon. Going across the table, the [6] M. Gerken.classes.uleth.ca/200501/chem2810a/lecture_4.pd effective nuclear charge increases because the electrons don f extracted 14/4/2014 Chemistry lecture notes. not move farther away from the nucleus (stay in the same [7] S. G. Abdu, 2010, Hartree-Fock solutions of the hydrogen, orbital). However, the charge of the nucleus increases as helium, lithium, beryllium and boron atoms. Nigerian more protons are present. Because of shielding, the effective Journal of Physics. 21(2). nuclear charge is somewhat less than the nuclear charge. [11] [8] F. Rioux, R. L. DeKock, 1998, J. Chem. Educ. 75, 537-539. Figure 5 shows the graphical representation of computed * effective nuclear charge Z1 against the experimental value [9] P. F. Lang, B. C. Smith, 2003, J. Chem. Educ. 8, 938-946. for a hydrogenic atom. From the figure it can be seen that as [10] wikianswers.com/Q/How_is_the_second_ionization_energy the nucleus becomes more positive, Z* increases. _different_from_the_first_ionisation_energyAccessed on 20/ The relationship between nuclear charge, Z and effective 2/2014. nuclear charge Z* is [11] www.xtremepapers.com/revision/a-level/chemistry/atoms/pr Z*=Z – S operties/moreies.php Accessed on 20/2/2014. where S is the shielding factor [12]. [12] www.chmdavidson.edu/ronutt/che115/zeff/zeff.htm Accessed on 8/3/2014.