Download Homework Set #1, Physics 570S, Theoretical Atomic, Molecular, and

Homework Set #1, Physics 570S, Theoretical Atomic, Molecular, and Optical Physics
Due in class on Friday, Sept.13
Problem 1. A reasonable model potential energy one can use to describe atomic electrons for the
lithium atom is given in atomic units by:
V(r)= Z(r)/r, where Z(r)= – 1 – 2 exp(– a r) –b r exp(– c r),
where the constants are given by: {a,b,c}={2.344, 0.1318, 1.422}. With this independent electron
model for the electrons in the lithium atom, calculate the following quantities. You may use
Mathematica, Matlab, Python, Fortran, etc., or other computer aids freely, but turn in your program as
part of your homework solution.
a) The binding energies of the 1s and 2s electrons in the lithium atom.
b) The absorption oscillator strength for the valence electron radiative transition 2s  2p, and the
spontaneous emission radiative lifetime of the 2p state in seconds.
c) The static polarizability of the Li(2s) state in atomic units, ignoring the contribution from the 1s
electrons, and assuming that only the 2s electron is affected by an external field. Compute this
polarizability two ways: first using second-order perturbation theory but including only the 2p
state in the infinite sum plus integral, and second using the Dalgarno-Lewis method.
d) Calculate and plot the photoionization cross section of the 2s state from the ionization threshold
up to photon energies of 20 eV.
Hint: A convenient way to carry out calculations of this type is to represent the unknown radial
solutions as a linear combination of a many (e.g. 15-30) known basis functions. This reduces a radial
homogeneous Schrödinger equation to a diagonalization problem (generalized eigenvalue problem), and
it reduces an inhomogeneous Schrödinger equation of the type that arises in the Dalgarno-Lewis
method to solution of an inhomogeneous linear system of equations. One such convenient choice of
basis set for calculations like this for orbital angular momentum L is the so-called Sturmian basis set,
defined as follows:
S(n,L,r)=Sqrt[n!/(2 (2L+1+n)!] Exp[-q r] (2 q r)L+1 Ln2L+1(2 q r)
(e.g., in Mathematica, the associated Laguerre function is given by Ln2L+1(2 q r) = LaguerreL[n,2L+1,2 q r] )
q is a constant that can be optimized for any given problem, and I believe a value around q=0.7 works
pretty well in this particular problem, though you might experiment with it. Since this is a
nonorthogonal basis set, one should be careful about reducing the differential equation(s) to
generalized eigenvalue problem(s) that can be solved numerically by matrix operations.