GerritJan Linker

CASSCF wavefunction for open shell singlet An open shell singlet is a state in which two electrons with opposite spin occupy a different orbital. It is an excited state compared to the singlet ground state in which both electrons occupy the same orbital. A ground state singlet wavefunction can be written as: Φ=a↑a↓, and the open shell singlet Φ=a↑b↓. In this notation Φ is the wavefunction a and b are the orbitals, ↑ and ↓ denote the spin function (either up or down). The orbitals are actually spin orbitals. A spin orbital is the product of a and b as their spatial component and the up and down as the spin component. Wavefunctions are commonly built as Slater determinants so they fulfill the Pauli principle, it can be expanded as follows: Φ=a↑b↓= a↑_{1}b↓_{2}  b↓_{1}a↑_{2}, where 1 and 2 denote the coordinates of electron 1 and 2. In CASSCF an open shell singlet can be constructed using 2 electrons in 2 orbitals: a CAS (2,2). In such a calculation the wavefunction can be built from 3 configuration state functions (CSF's) namely: Φ_{20}, Φ_{ud}, Φ_{02}. In this notation the 0,u,d,2 denote the occupation of the basis function with respectively 0, 1, 1, or 2. u and d denote the spin coupling, either up or down. To bring the two notations together, Φ_{20} can be written as Φ_{20}=a↑a↓, Φ_{ud} as Φ_{ud}=a↑b↓ or Φ_{ud}=b↑a↓ and Φ_{02} as Φ_{02}=b↑b↓ A open shell singlet wavefunction can be built from these 3 CSF's as: Ψ=c_{1}Φ_{ud} or Ψ=c_{2}Φ_{20}c_{3}Φ_{02} or written differently: Ψ=c_{1}a↑b↓ or Ψ=c_{2}l↑l↑c_{3}r↓r↓ Note that different sets of orbitals are used for the two wavefunctions. We can write the open shell singlet as the following linear combination Ψ=a↑b↓=a↑b↓a↓b↑ and then perform a transformation of the orbitals to be able to do this. So, in the [ab] basis the open shell singlet can be written as Ψ=a↑b↓a↓b↑ and a transformation exist of the basis [ab] to a basis [lr] so that we can write the open shell singlet as Ψ=c_{2}l↑l↑c_{3}r↓r↓ To show that the open shell singlet wavefunction can be built in this way we use the following transformation of the orbitals a and b to the orbitals l and r: l=(a+b)½√2, r=(ab)½√2 This leads to: [1] a=(l+r)/√2, b=(lr)/√2 We use the following wavefunction for the open shell singlet: [2] Ψ=a↑b↓a↓b↑ Expanding the determinants in [2] gives: [3] Ψ= ( (a↑_{1}b↓_{2}  b↓_{1}a↑_{2})  (a↓_{1}b↑_{2}  b↑_{1}a↓_{2}) ) = ( a↑_{1}b↓_{2}  b↓_{1}a↑_{2}  a↓_{1}b↑_{2} + b↑_{1}a↓_{2} ) Using [1] we rewrite [3] as: Ψ = ( (l↑_{1}+r↑_{1}) (l↓_{2}r↓_{2})/2  (l↓_{1}r↓_{1}) (l↑_{2}+r↑_{2})/2  (l↓_{1}+r↓_{1})(l↑_{2}r↑_{2})/2 + (l↑_{1}r↑_{1})(l↓_{2}+r↓_{2})/2 ) = ½( (l↑_{1}+r↑_{1}) (l↓_{2}r↓_{2})  (l↓_{1}r↓_{1}) (l↑_{2}+r↑_{2})  (l↓_{1}+r↓_{1})(l↑_{2}r↑_{2}) + (l↑_{1}r↑_{1})(l↓_{2}+r↓_{2}) ) expanding gives: [4] Ψ = ½( ( (l↑_{1}l↓_{2}+r↑_{1}l↓_{2})  (l↑_{1}r↓_{2}+r↑_{1}r↓_{2}) )  ( (l↓_{1}l↑_{2}r↓_{1}l↑_{2}) + (l↓_{1}r↑_{2}r↓_{1}r↑_{2}) )  ( (l↓_{1}l↑_{2}+r↓_{1}l↑_{2})  (l↓_{1}r↑_{2}+r↓_{1}r↑_{2}) ) + ( (l↑_{1}l↓_{2}r↑_{1}l↓_{2}) + (l↑_{1}r↓_{2}r↑_{1}r↓_{2}) ) ) = ½( l↑_{1}l↓_{2} + r↑_{1}l↓_{2}  l↑_{1}r↓_{2}  r↑_{1}r↓_{2}  l↓_{1}l↑_{2} + r↓_{1}l↑_{2}  l↓_{1}r↑_{2} + r↓_{1}r↑_{2}  l↓_{1}l↑_{2}  r↓_{1}l↑_{2} + l↓_{1}r↑_{2} + r↓_{1}r↑_{2} + l↑_{1}l↓_{2}  r↑_{1}l↓_{2} + l↑_{1}r↓_{2}  r↑_{1}r↓_{2} ) = ½( l↑_{1}l↓_{2} + l↑_{1}l↓_{2}  l↓_{1}l↑_{2}  l↓_{1}l↑_{2}  l↓_{1}r↑_{2} + l↓_{1}r↑_{2} + l↑_{1}r↓_{2}  l↑_{1}r↓_{2} + r↓_{1}l↑_{2}  r↓_{1}l↑_{2}  r↑_{1}l↓_{2} + r↑_{1}l↓_{2}  r↑_{1}r↓_{2}  r↑_{1}r↓_{2} + r↓_{1}r↑_{2} + r↓_{1}r↑_{2} ) = ½( 2l↑_{1}l↓_{2}  2l↓_{1}l↑_{2}  2r↑_{1}r↓_{2} + 2r↓_{1}r↑_{2} ) = ( l↑_{1}l↓_{2}  l↓_{1}l↑_{2}  r↑_{1}r↓_{2} + r↓_{1}r↑_{2} ) = ( (l↑_{1}l↓_{2}  l↓_{1}l↑_{2} ) (r↑_{1}r↓_{2}  r↓_{1}r↑_{2} )) = l↑l↓  r↑r↓
