The approximate ground-state LCAO electronic configuration of the H2 molecule is (σg1s) 2 . Write down the corresponding approximate two-electron eigenfunction corresponding to this configuration and the associated Slater determinant that produces this eigenfunction. b. (5 points) Now consider an excited state in which one of the electrons is the next bonding orbital, which is the σg2s orbital. The electronic configuration is (σg1s) 1 (σg2s) 1 . Assuming that the electrons do not change their spin state from that of the ground state in part a, write down the corresponding approximate two-electron eigenfunction. c. (20 points) The approximate excited-state eigenfunction you wrote down in part b cannot be written as a single Slater determinant, but it can be written as a sum or difference of two Slater determinants. Find these two determinants and determine if you need a sum or difference of the two to obtain the approximate excited-state eigenfunction.

Linear combination of atomic orbitals (LCAO) is a simple method of quantum chemistry that yields a qualitative picture of the molecular orbitals (MOs) in a molecule. Let us consider H

+

2

again. The approximation embodied in the LCAO approach is based on the notion that when the two protons are very far apart, the electron in its ground state will be a 1s orbital of one of the protons. Of course, we do not know which one, so we end up with a Schrödinger cat-like state in which it has some probability to be on one or the other.

As with the HF method, we propose a guess of the true wave function for the electron

ψg(r)=CAψ

A

1s

(r)+CBψ

B

1s

(r)

where ψ

A

1s

(r)=ψ1s(r−RA) is a 1s hydrogen orbital centered on proton A and ψ

B

1s

(r)=ψ1s(r−RB) is a 1s hydrogen orbital centered on proton B. Recall ψ1s(r)=ψ100(r,ϕ,θ). The positions RA and RB are given simply by the vectors

RA=(0,0,R/2)RB=(0,0,−R/2)

The explicit forms of ψ

A

1s

(r) and ψ

B

1s

(r) are

ψ

A

1s

(r)=

1

(πa

3

0

)1/2

e−|r−RA|/a0ψ

B

1s

(r)=

1

(πa

3

0

)1/2

e−|r−RB|/a0

Now, unlike the HF approach, in which we try to optimize the shape of the orbitals themselves, in the LCAO approach, the shape of the ψ1s orbital is already given. What we try to optimize here are the coefficients CA and CB that determine the amplitude for the electron to be found on proton A or proton B.

Explanation:

the mantle

explanation:

earth is divided into three layers: the crust or the outermost layer on which we live; the underlying, middle layer - the mantle - and the core (the innermost layer of earth).