### Author Topic: ADC(2) transition moment calculation  (Read 3411 times)

#### david.mester

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• Karma: +0/-0 « on: March 22, 2017, 07:55:39 am »
Dear Users and Developers,

I would like to perform ADC(2) transition moment calculations with Turbomole and make comparisons with previous results and benchmarks obtained with other programs.

Could you, please, help to clarify the precise expression for the transition moments printed by Tubromole? The corresponding section of the manual (http://www.turbomole-gmbh.com/manuals/version_6_6/Documentation_html/DOKse43.html) discusses that some terms are neglected in the ground to excited state transition moments:

...the implementation in the ricc2 program neglects in the calculation of the ground to excited state transition moments the contributions which are second order in ground state amplitudes (i.e. contain second-order amplitudes or products of first-order amplitudes)

It is not clear to us, which contributions are included and which ones are neglected. Could you, please, explain which terms are computed in more detail, because we were not able to find a more detailed reference/publication covering the implementation in Turbomole. Could you point us to such documentation in the literature that we perhaps missed?

To assist the discussion I collected here the terms contributing to the ADC(2) transition densities according to the Eqs. (A1)-(A3) in the paper of Dreuw and co-workers (Mol. Phys., 2014, 112, 774-784):

0th order:
\rho_{ai} = Y_{ia}
1st order:
\rho_{ia} = - \sum_{jb} t_{ij}^{ab} Y_{jb}
2nd order:
\rho_{ij} = - \sum_{a} \rho^{MP2}_{ia} Y_{ja} - \sum_{kab} Y_{ik}^{ab} t_{jk}^{ab}
\rho_{ia} = - \sum_{jb} Y_{jb} t^D_{ij}^{ab}
\rho_{ai} = 1/2 \sum_{jb} t_{ij}^{ab} \sum_{kc} t_{jk}^{bc} Y_{kc} - 1/2 \sum_{b} \rho^{MP2}_{ab} Y_{ib} + 1/2 \sum_{j} \rho^{MP2}_{ij} Y_{ja}
\rho_{ab} = \sum_{i} Y_{ia} \rho^{MP2}_{ib} + \sum_{ijc} Y_{ij}^{ac} t_{ij}^{bc},

where Y denotes the excited state eigenvectors, t_{ij}^{ab} is the MP2 amplitude, t^D_{ij}^{ab} is an O(N^6) scaled intermediate, and \rho^{MP2}_{pq} is the pq part of the MP2 density matrix.

Could you explain your approximation using the above terminology?

An second source for discrepancies in comparisons could come from different approaches to normalize the ground/excited state wave function. Could you explain how do you normalize these functions in Tubromole?

Yours sincerely,
Dávid Mester
« Last Edit: March 22, 2017, 07:59:08 am by david.mester »

#### christof.haettig

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• Karma: +0/-0 ##### Re: ADC(2) transition moment calculation
« Reply #1 on: March 23, 2017, 02:58:20 pm »
I hope you do not expect me to analyze the notation from Dreuw....

The implementation in TURBOMOLE is based on the original formulation by Schirmer and coworker. They used a strictly MP-based approximation which
leads to the occurence of second-order contributions in the expressions for transition moments. That might be interesting for academic purposes,
but for a model that could be used for routine (production) calculations it is for the performance/cost ratio unacceptable to include the O(N^6) scaling
terms second-order terms.
There identification is straightforward if one notes that a) the MP2 energy is computed from the first-order amplitudes and the b) the second-order
amplitudes are those computed in a MPPT program to compute the MP4 energy.
The detailed working expressions are given e.g. by Lunkenheimer in JCTC 9 (2013) 977.
« Last Edit: March 24, 2017, 03:43:11 pm by christof.haettig »