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Configuration Interaction and Performance of various levels of electronic structure theory for H2.
Typology: Lecture notes
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Following^ the^ MO^ treatment^ of^ H2+,
assume^ the^ (normalized)^ ground
electronic state wavefunction is given by:state^ wavefunction^ is^ given^ by:^ [^ (1)^ (2)^ (1)^ (2)^ (2)^ (1) /^2 ]
(^ ,^ ) s s s s r rA B A B
1 (2)^1 (2)^ (^ ,^ ) s s s^ s^ r rA B A^ B 2(1 ) (^) S
Evaluate^ the^ ground^ state^ electronic
energy^ based^ on^ this^ presumed (approximate) eigenfunction:(approximate)^ eigenfunction:
0 1 2 1 2 1 2 1
2 '^2 '^2 ( ,^ )^ ( ,^ )^2
1 2(1^ ) gs^ el^
j^ j^ k^ j^ k^ m^ s l E^ dr dr^ r r^ H^ r r^
=^ =^
+^ −^ ++^ +
1 2(1^ ) R S^ S +^ +
Spin‐orbitals^ of^ type^1 and^3 have
the^ same^ symmetry,^ and^ therefore can^ “mix”^ (to^ give^ improved^ wavefunctions
and^ energy^ eigenvalues):(1) (2) (1) (2) (2) (1) /^2 ψ ψ α β β α [ ] 1 + +
ψ^ ψ^ α^ β^ β^ αΨ = −+^ + Consider the trial wavefunction: c^ c Ψ =^ Ψ +^ Ψ^ [1]^1 1 2 2 tr (^) Inserting this trial function^ into^ the^ Raleigh‐Ritz^ Variational
Principle^ … we^ will^ determine^ (c1,c2)^ that^ minimize
the^ expectation^ value^ of^ the Electronic^ Hamiltonian^ operator^ with^ this^ trial^ function.^ This^ will^ provide us^ with^ improved^ values^ of^ the^ ground
state^ energy^ (which^ must^ be^ lowered by this variational calculation) and the ground state energy eigenfunction.by^ this^ variational^ calculation)^ and
the^ ground^ state^ energy^ eigenfunction.
Note^ the^ flexibility^ of^ the^ trial^ (spatial)
wavefunction.^ Modulo^ normalization: (1)^ (2)^ (1)^ (2) c c ψ ψψ^ ψ+ 1 2 + +^ −^ − 1
2 [1^ (1)^1 (1)][1^ (2)^1 (2)]
[1^ (1)^1 (1)][1^ (2)^1
(2)] A^ B^ A^ B^
A^ B^ A^ B c^ s^ s^ s^ s^
c^ s^ s^ s^ s =^ +^ +^
+^ −^ − 1 1 1 (1)1^ (2) 1 2 1 (1)1^ (2) A B B^ A c^ c^ s^ s^
s^ s =^ = −^ ⇒^
Note^ the^ reductions^ for^ specific^ c1,c
coefficient^ choices:Covalent
bonding 1 ,^1 1 (1)1^ (2) 1 2 1 (1)1^ (2) A B B^ A c^ c^ s^ s^
s^ s ⇒ + 1,^1 1 (1)1^ (2) 1 2 1 (1)1^ (2) A A B^ B c^ c^ s^ s^
s^ s =^ =^ ⇒^ +
configurationIonic^ bondingconfigurationconfiguration Thus,^ the^ trial^ function^ ψ_tr in^ Eq.
[1]^ confers^ considerable^ additional flexibility^ to^ the^ wavefunction^ shape,
hence^ leading^ to^ a^ more^ accuratey p , g solution^ of^ the^ Schrodinger^ Eq.
Testing^ qualitative^ MO^ theory
prediction^ of^ Bond^ Order^ with
experiment g q^ y p^
p
for^ homonuclear^ diatomics made
st^ from elements in the 1 row^ of^ the^ Periodic Table^ (using^ the^ “Molecular^ Orbital
Aufbau”^ principle): Bond Order [# '^ #^ ' ] / 2 bonding e s^ antibonding e s ≡ − [D.A. McQuarrie, Quantum Chemistry ]
nd^ Going to the 2 row^ of^ the^ Periodic Table^ …^ need^ to^ build^ up^ MO’s nd^ from the 2Lewis Shell:from the 2 Lewis^ Shell: p_z orbitals^ can^ also^ be^ “added^ and subtracted”^ to^ form^ g=gerade and u=ungerade combinations:^ [D.A.^ McQuarrie,^ Quantum^ Chemistry
]
Qualitative^ MO^ theory^ orbital^ diagram
for^ homonuclear^ diatomics st^ nd^ composed of 1 or^2 row^ elements: difficult to distinguish the ordering of these MOs:g [D.A. McQuarrie,^ Quantum^ Chemistry ]
Testing^ qualitative^ MO^ theory^ prediction
of^ Bond^ Order^ with^ experiment for^ homonuclear^ diatomics made
st^ from elements in the 2 row^ of^ the Periodic Table:^ [D.A.^ McQuarrie,^ Quantum^ Chemistry
]
