# GW with fully analytic formula (general)

Here follows the most standard \(GW\) calculation with MOLGW for the ionization potential (IP) of water. \(G_0W_0\) based on BHLYP (50 %) of exact-exchange is known to be very good for IPs.

```
&molgw
comment='H2O GW analytic formula' ! an optional plain text here
scf='BHLYP'
basis='cc-pVTZ'
auxil_basis='cc-pVTZ-RI'
postscf='G0W0'
selfenergy_state_range=0 ! will calculate just the HOMO
frozencore='yes' ! accurate approximation: O1s will not be included
! in the RPA/GW calculation
natom=3
/
O 0.000000 0.000000 0.119262
H 0.000000 0.763239 -0.477047
H 0.000000 -0.763239 -0.477047
```

After the SCF cycles with the BHLYP hybrid functinal, one can find the response calculation within the RPA equation:

```
Prepare a polarizability spectral function with
Occupied states: 4
Virtual states: 53
Transition space: 212
```

MOLGW will then diagonalize a \(212 \times 212\) matrix. This is must often the computational bottleneck in a calculation. Memory scales as \(N^4\) and computer time as \(N^6\).

Then comes the \(GW\) quasiparticle energies and weights:

```
state spin QP energy (eV) QP spectral weight
5 1 -12.354761 0.915241
```

State 5 is the HOMO. The \(GW\) ionization potential is the negative HOMO, here 12.35 eV.

Experimental value is 12.62 eV

The discrepancy is large. Why?

## GW slow convergence

A complete basis convergnce study for GW@BHLYP would give

with a CBS HOMO evaluated to -12.72 eV.

MOLGW automatically proposes a CBS extrapolated value based a *trained* linear regression:

```
Extrapolation to CBS (eV)
<i|-\nabla^2/2|i> Delta E_i E_i(cc-pVTZ) E_i(CBS)
state 5 spin 1 : 62.271159 -0.404451 -12.354761 -12.759212
```

Now the experimental and \(GW\) IP agree within 0.14 eV. This is the typical accuracy of \(GW\) calculations.

# GW with imaginary frequencies (HOMO LUMO gap)

If we are just interested in the HOMO LUMO region, there exists an alternative to the fully diagonalization of the RPA equation (\(N^6\) scaling).

The \(GW\) self-energy can be evaluated by numerical quadruture for imaginary frequencies and then analytically continued to real frequencies with a PadÃ© approximant.

This is much faster (\(N^4\) scaling), however it is robust only in the HOMO-LUMO gap region and it requires more convergence parameters.

Here is a typical input file:

```
&molgw
comment='H2O GW with numerical integration'
scf='BHLYP'
basis='cc-pVTZ'
auxil_basis='cc-pVTZ-RI'
postscf='G0W0_PADE'
selfenergy_state_range=0
frozencore='yes'
nomega_chi_imag=16 ! frequency grid for the response function
nomega_sigma_calc=12 ! frequency grid for sigma along the imaginary axis
! (fit of the analytic continuation)
step_sigma_calc=0.1
nomega_sigma=101 ! frequency grid for sigma along the real axis
natom=3
/
O 0.000000 0.000000 0.119262
H 0.000000 0.763239 -0.477047
H 0.000000 -0.763239 -0.477047
```

The output reports a HOMO value of -12.354 eV, which agrees within 1 meV with the previous analytic value.

```
state spin QP energy (eV) QP spectral weight
5 1 -12.354356 0.915308
```