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Question 3. Qualitatively explain what is meant by the equation:
|φi = X|iihi|φi
alli
In your explanation, be sure to include the qualitative interpretations of hi|φi, state vector |φi, state vectors |ii, and what we mean by the term “resolution of the identity”.
Question 4. Qualitatively (i.e., without equations, unless you really prefer it) explain the major steps required to get from the time-independent Schrodinger equation to the Roothaan-Hall equations. Make sure to touch on the following details:
• the Hamiltonian
• the form and composition of the wavefunction
• how the expectation value of the Hamiltonian for this wavefunction E = hΦ | Hˆ | Φi relates to the Hartree-Fock equations
• how the Hartree-Fock equations relate to the Roothaan-Hall equations
Question 5. Beginning from the restricted-determinant form for the 1st Slater-Condon rule:
N/2 N/2N/2
E = 2Xhφi|hˆ|φii + XX2hφiφj|gˆ|φiφji − hφiφj|gˆ|φjφii
i i j
expand each spatial orbital as a linear combination of atomic orbitals:
hφi |= Xhχp | Cpi∗ | φii = X | χqiCqi hφj |= Xhχr | Crj∗ | φji = X | χsiCsj
p q r s
and derive the RHF energy expression for molecular orbitals (MOs) approximated by a linear combination of atomic orbital basis functions.
Question 6. Suppose you have a basis-set composed of 4 Gaussian basis functions
χ1(r) χ2(r) χ3(r) χ4(r)
Suppose further you perform a restricted Hartree-Fock computation (i.e., using the Roothaan-Hall equations) using this basis on a system with 4 electrons and obtain the following molecular orbital coefficients:
C
where each column above corresponds to the coefficients for one molecular orbital, and the relative energies of each molecular orbital are in increasing order from left to right.
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