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Supreeta N.
In this problem you will study the ground state energy of atoms (or ions) with two electrons using perturbation theory. Let us consider an atom or ion consisting of a nucleus of charge Ze and two electrons of charge -e. If Z = 1 the system is a negative hydrogen ion (H-), if Z = 2 it is a helium atom (He), if Z = 3 a positively charged lithium ion (Li+), and so on. The Hamiltonian of the system is then (assuming that the mass of the nucleus is infinitely large as compared to the mass of the electrons): H = -(h^2/2m)∇_1^2 - (h^2/2m)∇_2^2 - Ze^2/(4πε_0r_1) - Ze^2/(4πε_0r_2) + e^2/(4πε_0r_12) where r_12 = |r_1 - r_2|. Because of the presence of the electron-electron interaction term e^2/(4πε_0r_12) in the Hamiltonian the corresponding Schrödinger equation cannot be solved exactly. Investigate the problem with the help of time-independent perturbation theory by treating the electron-electron interaction term as perturbation H'. a) Determine the normalized ground state wave functions Ψ_0(r_1, r_2) and corresponding ground state eigenenergy E_0^(0) of the Schrödinger equation for the unperturbed Hamiltonian H_0 = H - H' as functions of the charge Z. Is the ground state energy level degenerate or non-degenerate? b) Using first-order perturbation theory determine the first-order correction to the ground state energy, E_0^(1), as a function of Z. c) Calculate the ground state energy up to first-order correction, E_0 = E_0^(0) + E_0^(1), for the two-electron atoms and ions up to Z = 6. Determine the relative error of your results as compared to the accurate results (H-: -14.4 eV, He: -79 eV, Li+: -198.1 eV, Be^2+: -371.7 eV, B^3+: -599.5 eV, C^4+: -881.9 eV). Does the relative error increase or decrease with increasing Z? Give a physical interpretation of this trend.
Adi S.
Allowed values for the quantum numbers of electrons are as follows: n = 1, 2, 3, ... l = 0, 1, 2, 3, ..., n - 1 ml = 0, ±1, ±2, ±3, ..., ±l ms = ±1/2 The relationships between n and the shell designations are noted in Table 2.1. Relative to the subshells, l = 0 corresponds to an s subshell l = 1 corresponds to a p subshell l = 2 corresponds to a d subshell l = 3 corresponds to an f subshell For the K shell, the four quantum numbers for each of the two electrons in the 1s state, in the order nlmlms are 100 (1/2) and 100 (-1/2). Write the four quantum numbers for all of the electrons in the L and M shells, and note which correspond to the s, p, and d subshells. Without consulting Figure 2.6 or Table 2.2, determine whether each of the electron configurations given below is an inert gas, a halogen, an alkali metal, an alkaline earth metal, or a transition metal. Justify your choices. (a) 1s²2s²2p⁶3s²3p⁶3d⁷4s² (b) 1s²2s²2p⁶3s²3p⁶ (c) 1s²2s²2p⁵ (d) 1s²2s²2p⁶3s² (e) 1s²2s²2p⁶3s²3p⁶3d²4s² (f) 1s²2s²2p⁶3s²3p⁶4s¹ 4. Calculate the bonding energy E₀ in terms of the parameters A, B, and n using the following procedure: I. Differentiate EN with respect to r, and then set the resulting expression equal to zero, because the curve of EN versus r is a minimum at E₀. II. Solve for r in terms of A, B, and n, which yields r₀, the equilibrium interionic spacing. III. Determine the expression for E₀ by substituting r₀ into the above equation.
Sri K.
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