Home work Physical and Theoretical Chemistry I II
- Structure of Matter -

Discussion Fr 10.6.2005 at 10:30am lecture hall SN 20.2

Home work 7 (LCAO, harmonic oscillator, HCl)

Exercise 1:
During the lecture we calculated the energies EI and EII of a system given by two configurations |1> and |2>, where both configurations were equivalent, i.e. they are described by the same energy Eo = <1|H|1> = <2|H|2> . Now calculate the energy under the assumption that the energies are different, i.e. E1 = <1|H|1> and E2 = <2|H|2> with E1¹E2.

a) Show that in case of a small interaction energy (H21 << E1,E2 ) between |1> and |2> the energetically lower level E1 is shifted to lower energies while the energetically higher level E2 is shifted to higher energies by about the same amount ΔEw.
b) Show that the coefficients a1I and a2I, which describe the resulting lower energy level EI, |I> = a1I |1> + a2I |2>, are given by:
a1I/a2I = (E1−E2)/H21 | I > = {H21² + (E1−E2)²}−½ · [(E1−E2) | 1 > + H21 | 2 >]
We used the abbreviation H12 = <1|H|2> and H12=H21.

Exercise 2:
Show that the commutator of the creation a+ = 1/Ö2(−d/dz + z) and the annihilation operator a- = 1/Ö2(d/dz + z) is given by [a+,a-] = -1.

Exercise 3:
The wavefunction of the harmonic oscillator for the ground state and the first excited state are given by :
Y 0(z) = c0 e−z²/2 and Y 1(z) = c1 z e−z²/2 , respectively. z² = x² (mw/h)½
a) Determine the constants c0 and c1 by using: -¥ò+∞e−az²dz = (π/a)½ and -¥ò+∞ z2 e−az²dz = ½ (π/a3)½
b) What is the parity of the wavefunctions Y0(z) and Y1(z).
Parity characterizes the system under the symmetry operation of inversion. For a one dimensional wavefunction Y0(z) and Y1(z) it is the replacement of z by -z. The parity of a wavefunction is called gerade (g), if Yg(-z) = + Yg(z), and ungerade (u), if Yu(-z) = − Yu(z).

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# Exercise 4:

a) The distance between two neighboring levels of vibration in the HCl molecule is 2991 cm-1, where 1 cm-1 = 1.986·10-23 J. What is the value of the zero point energy ?
b) Determine the maximum relativ velocity of the atoms in the HCl molecule in the vibrational ground state v = 0.
Use the reduced mass of HCl mHCl = mHmCl /(mH+mCl) with mCl = 35 u.
c) Determine the value of the force constant k.
d) What is the zero point energy and the distance between neighboring vibrational states in the DCl molekule ?