Problem Set 10 Solutions

Transcription

1 Chemisty 6 D. Jean M. Standad Poblem Set 0 Solutions. Give the explicit fom of the Hamiltonian opeato (in atomic units) fo the lithium atom. You expession should not include any summations (expand them out). The geneal fom of the Hamiltonian opeato (in atomic units) fo an atom is ˆ H = ˆ i i= Z i=, i= j =i ij whee the "del squaed" opeatos defined as ˆ i = x i y i z. i Fo Li, thee ae thee electons (=) and the atomic numbe Z=, so the opeato becomes Ĥ = ˆ i i= i=. i= j=i ij Witing out each tem in the opeato explicitly, we have Ĥ = ˆ ˆ ˆ. The fist thee tems ae the kinetic enegy opeatos fo electons,, and, while the fouth though sixth tems ae the electon-nuclea attactions (fo the thee electons), and the last thee tems ae the electonelecton epulsions.

2 . Give the explicit fom of the Hamiltonian opeato (in atomic units) fo the beyllium atom. You expession should not include any summations. Classify the vaious tems that ae included in the Hamiltonian opeato. The geneal fom of the Hamiltonian opeato (in atomic units) fo an atom is ˆ H = ˆ i i= Z i=, i= j =i ij whee the "del squaed" opeatos defined as ˆ i = x i y i z. i Fo Be, thee ae fou electons (=) and the atomic numbe Z=, so the opeato becomes ˆ H = ˆ i i= i=. i= j =i ij Witing out each tem in the opeato explicitly, we have H ˆ = ˆ ˆ ˆ ˆ. The fist fou tems ae the kinetic enegy opeatos fo electons,,, and, while tems 5-8 ae the electonnuclea attactions, and the last six tems (on the second line) coespond to the electon-electon epulsions.

3 . Fo a Li atom with electon configuation s s, list the possible sets of quantum numbes fo electons,, and. Make sue to include all five quantum numbes, n, l, m l, s, and m s fo each electon. How many diffeent sets of quantum numbes ae possible? We must apply the Pauli Pinciple to these electons. Two of the electons (labeled and below) occupy the same s obital, so we must make sue that they have diffeent sets of quantum numbes. The table below lists the possible quantum numbes fo the two s electons. numbe n l 0 0 m l 0 0 s / / m s / / ote that all the quantum numbes ae equied to be the same because both electons occupy the s obital except the m s quantum numbes, which have to be diffeent fo the two electons in ode to satisfy the Pauli Pinciple. is the only electon in the s obital, so the diffeence in the pincipal quantum numbe gives it a unique set of quantum numbes when compaed with the s electons. Thee ae two possibilities fo the quantum numbes of electon : numbe Option Option n l 0 0 m l 0 0 s / / m s / / This electon can be eithe spin up o spin down. Theefoe, thee ae two possible sets of quantum numbes fo the electons in Li. These ae listed in the table below. The two sets coespond to the same choice fo the two s electons (the only option) and eithe option o option fo the s electon. Set n l m l s / / / m s / / / numbe

4 . Continued Set numbe n l m l s / / / m s / / /

5 5. Conside the spin functions given by the following elations: f (,) = α() α()α() α() g(,) = α() β () α() β () u(, ) = α() α() υ(,) = α() β () β () β (). Detemine which of the functions ae antisymmetic with espect to intechange of the electon coodinates. Also detemine which of the functions ae symmetic with espect to intechange of the electon coodinates. Fo the function f (, ), the intechange of electons and gives f intechange (, ) = α() α()α() α() = f (, ). Intechanging electon coodinates in the function f (, ) yields the same function. Thus, this function is symmetic with espect to intechange. Fo the function g(, ), the intechange of electons and gives g intechange (, ) = α()β() α()β() = g(, ). Intechanging electon coodinates in the function g(, ) yields the same function. Thus, this function is symmetic with espect to intechange. Fo the function u(, ), the intechange of electons and gives u intechange (, ) = α() α() = u(, ). Intechanging electon coodinates in the function u(, ) yields times the function. Thus, this function is antisymmetic with espect to intechange. Fo the function υ(, ), the intechange of electons and gives υ intechange (, ) = α()β() β()β() ± υ(, ). Thus, the function υ(, ) is neithe symmetic no antisymmetic with espect to intechange.

6 6 5. A eseache employs the following function to epesent the spatial pat of the gound state wavefunction of the Li atom, ψ(,,) = χ s ( ) χ s ( ) χ s ( ). Hee, χ s o χ s is shothand notation used to efe to the spatial fom of the atomic obital; fo example, χ s ( ) = $ & % πa o ' ) ( / e / a o. Does the fom given above fo the Li atom coespond to an acceptable spatial wavefunction? Explain. o, it is not an acceptable function. The electon configuation of the gound state of Li is s s, so the atomic obitals used to constuct the function ae coect. Howeve, fo example, the function specifically places electon in the s obital; thus, the function distinguishes between the electons. An acceptable function would allow fo electon,, o being placed in the s obital. Shown below is an example of an acceptable spatial wavefunction that does not distinguish between the electons: ψ(,,) = [ χ s ( ) χ s ( ) χ s ( ) χ s ( ) χ s ( ) χ s ( ) χ s ( ) χ s ( ) χ s ( ) ]. This function places each of the thee electons in the s obital, so it does not distinguish. This example spatial wavefunction happens to be symmetic with espect to intechange of electons. When constucting the full wavefunction fo Li, including spatial and spin components, this symmetic spatial wavefunction would have to be combined with an antisymmetic spin wavefunction so that the oveall wavefunction was antisymmetic and satisfied the Pauli Pinciple.

7 7 6. Use the Pauli and Aufbau pinciples, along with Hund's Rule of maximum multiplicity, to daw obital enegy diagams fo the gound states of the Si and P atoms. Classify the multiplicity of the gound states of the Si and P atoms as singlet, doublet, tiplet, o quatet. Using the Pauli and Aufbau Pinciples, the electon configuation of Si is s s p 6 s p. The electon configuation of P is s s p 6 s p. Obital enegy diagams fo Si and P ae shown below. ote that using Hund's ule of maximum multiplicity, the electons in the p obitals ae placed in diffeent obitals and have the same electon spin. Atomic Enegy Levels of Si Atomic Enegy Levels of P p p s s E p E p s s s s Fo Si, placing two electons in the p set of obitals leads to two unpaied electons. Thus, the total spin of the two unpaied electons is S=, and theefoe the multiplicity is S = =. Theefoe, the multiplicity of the gound state of Si is a tiplet. Fo P, placing thee electons in the p set of obitals leads to thee unpaied electons. Thus, the total spin of the thee unpaied electons is S=/, and theefoe the multiplicity is S = (/) =. Theefoe, the multiplicity of the gound state of P is a quatet.