Thanks a lot for your fast and extensive answers. I am still working on my problem but here is an update:

The numbering is clear to me now. I just assume with spin “alpha” and “beta” you refer to spin up or down, so that part is fine now.

I am still going through the book chapter, which is a great reference. I have seen factors of 1, 1/2 or 1/4 in front of the sum in the T2 definitions: T2 = 1/4 Sum_{ijab}t_{abij} a_a^daggger a_b^dagger a_i a_j

In this case the sum is unrestricted so that is where the 1/4 comes from (double counting) and other people write the sum without double counting and hence no prefactor…

My goal is numerically calculate the wavefunction that has the same energy as the cc calculation outputs. Therefore I am trying to understand how to achieve this given the cc amplitudes. (My ultimate goal is to represent this wavefunction on a quantum computer, but that step I will tackle once I can prepare the cc wavefunction classically)

For H2 using 4 basis functions:

The only cc amplitude is

TIjAb Amplitudes:

0 0 0 0 -0.1153404926

I need to work in the spin orbital picture so the hartree fock state is |HF> = | 0 0 1 1> where the lowest two spin orbitals are occupied. In this wavefunction the spin orbitals are ordered from right to left:

|1. spatial orbital spin down, 1. spatial orbital spin up, 0. spatial orbital spin down, 0. spatial orbital spin up>

In my spin orbital numbering below:

0 is 0. spatial orbital spin up (occupied)

1 is 0. spatial orbital spin down (occupied)

2 is 1. spatial orbital spin up (unoccupied)

3 is 1. spatial orbital spin down (unoccupied)

In this case TIjAb = 0 0 0 0 refers to an operator

a_0 a_1 a_2^dagger a_3^dagger

Is this correct?

After a lot trying I constructed the wavefunction

|CC> = exp(0.1153404926 * a_0 a_1 a_2^dagger a_3^dagger) | 0 0 1 1>

Not that I changed the sign of the cc amplitude (why do I need to do that?) and obviously this wavefunction is not normalized anymore. But if I compute the energy and normalize this wavefunction:

average energy = < CC | Hamiltonian | CC > / <CC | CC>

Then I get exactly the same energy as the coupled cluster calculation gave me.

Great. Except for the minus sign everything seems to work but this is not the case when looking at H4 using 8 basis function I get the following cc amplitudes:

TIA Amplitudes:

0 0 -0.0062919007

1 1 0.0030948932

Tia Amplitudes:

TIJAB Amplitudes:

Tijab Amplitudes:

TIjAb Amplitudes:

1 1 0 0 -0.1299315906

0 0 0 0 -0.049109294

0 1 0 1 -0.0468249907

1 0 1 0 -0.0468249907

0 0 1 1 -0.029138475

1 1 1 1 -0.0272126304

0 1 1 0 -0.0240235256

1 0 0 1 -0.0240235256

In this case I didn’t manage so far to construct a wavefunction as in the case above for H2 which has the same energy as the cc calculation. Maybe I am making a mistake or there are signs which I do wrong.

Again I would use the Hartree Fock state | 0 0 0 0 1 1 1 1> and try to apply exp(T) | HF> in order to get a wavefunction:

Again I am using my spin orbital numbering:

In my spin orbital numbering below:

0 is 0. spatial orbital spin up (occupied)

1 is 0. spatial orbital spin down (occupied)

2 is 1. spatial orbital spin up (occupied)

3 is 1. spatial orbital spin down (occupied)

4 is 2. spatial orbital spin up (unoccupied)

5 is 2. spatial orbital spin down (unoccupied)

6 is 3. spatial orbital spin up (unoccupied)

7 is 3. spatial orbital spin down (unoccupied)

A term

TIA Amplitudes:

0 0 -0.0062919007

in my spin orbital notation refers to -0.0062919007 a_0 a_4^dagger so I would add this term to T. Is this the correct sign or is there again a minus sign coming from somewhere?

Do I also have to add -0.0062919007 a_1 a_5^dagger due to symmetry reasons?

Same question arise for the TIjAb terms as I need to understand what exactly the T operator is with the correct signs for each term…

Also I am not sure if this way of generating a wavefunction and then normalizing it should give me a wavefunction which has the energy equal to what the cc calculation outputs…