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Re: relaxed/unrelaxed density excited state properties

Alex Granovsky

Dear Sanya,

Dealing with the wavefunctions that are the exact solutions
of Schroedinger equation, one can uniquely define one-body
density matrix using pretty simple formulas of Quantum Mechanics.

However, Quantum Chemistry is quite a different case :-)
Here we are forced to use incomplete basis sets and various
approximate methods...

There are computational methods that do have (approximate)
wavefunctions, e.g., HF, CI, MCSCF. Here, to have wavefunction
means that the expressions for energy used within these methods
are exactly equivalent to the expectation value of Hamiltonian
operator calculated using those approximate wavefunctions.

Some other methods do not have (even approximate!) wavefunctions
at all, e.g., DFT; TDHF and TDDFT; MP2, MP3, MP4,...,MPn; CCSD and
any other CC-like methods;  EOM-CC etc... This means that the
approximate wavefunction either does not exist at all
(e.g. in DFT, TDHF and TDDFT), or, alternatively, the energy
evaluated within the framework of these methods using some
approximation to "wavefunction", is not equal to the expectation
value of Hamiltonian calculated using this "wavefunction" (e.g.,
all MPn and CC-like methods).  

The methods that have approximate wavefunction also have well-defined
expectation-value type one-electron density matrix.
It is defined using standard definition of Quantum Mechanics.
This density matrix has lots of interesting properties, e.g.,
occupation numbers are bracketed between 0 and 2 (for the total
density matrix). The expectation value of any one-electron property
is simply the trace of density matrix multiplied on the operator
representing this property.

However, even in this simplest case, it is possible to define
density matrix of another type - the so-called response-type
density matrix. It is defined so that the derivative of energy
with respect to any one-electron perturbation (x*V) is equal
to the trace of this density matrix times V:

-- = tr(D*V)

These two definitions are identical for exact solutions of Schroedinger
equation. However, for approximate solutions (e.g. working
with incomplete basis sets that is typical for QC) they differs.
More precisely, for those methods where the energy is fully
optimized with respect to molecular orbitals (HF, MCSCF,
complete CI) they are identical, otherwise they are different.
For example, for any incomplete CI (e.g. CISD) one can define both
expectation value and response-type density matrices. One can
calculate dipole moments using these two density matrices and
get two different answer - the first one is the expectation
value of dipole operator, the second one is the derivative of
CISD energy with respect to the applied electric field.

The response-type density does not have many useful properties of
the "real", expectation value density matrix. For example,
occupation numbers can be arbitrary. However, it does not require
any wavefunction to be defined! One just need to obtain expression
for derivative of total energy with respect to arbitrary one-particle
perturbation - and this can be done for virtually any computational
method, including DFT, MPn, any CC-like, EOM-CC and TD-HF/TD-DFT.
This density matrix is exactly what is usually called "relaxed
density". Typically, it naturally becomes available as the
byproduct during molecular gradients calculations.

Finally, one can define some approximations for density matrix,
even for methods without wavefunction. E.g. for MP2, one can
define approximation to the true density matrix that is correct
up to the second order of PT. For TDHF and TDDFT, one can define
approximation that resembles expectation value density matrix for
CI singles (CIS). This approximation (and there are some good
reasons for this) is what is called "unrelaxed density".

To summarize, one cannot directly compare dipole moments calculated
using relaxed and unrelaxed densities. However, with Firefly, one
can calculate dipole moment as the derivative of energy wrt. to
electric field using runtyp=ffield. This quantity can be directly
compared with dipole moment calculated using relaxed density.

Hope this helps.

Alex Granovsky

On Tue Sep 22 '09 6:49pm, sanya wrote
>Dear All,

>When calculating excited state properties, such as dipole moments etc., some programs give "relaxed density excited state properties", while others give "unrelaxed density excited state properties". What is the difference between them? As far as I understand, Firefly calculates properties using unrelaxed density (at least, in TDHF and TDDFT). Is it correct to compare, say, dipole moments calculated using relaxed and unrelaxed densities calculated by different programs (with the same functional and basis set, of course)?

[ This message was edited on Sun Oct 4 '09 at 11:24pm by the author ]

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