In quantum chemistry, n-electron valence state perturbation theory (NEVPT) is a perturbative treatment applicable to multireference CASCI-type wavefunctions. It can be considered as a generalization of the well-known second-order Møller–Plesset perturbation theory to multireference Complete Active Space cases. The theory is directly integrated into many quantum chemistry packages such as MOLCAS, Molpro, DALTON, PySCF and ORCA.
The research performed into the development of this theory led to various implementations. The theory here presented refers to the deployment for the Single-State NEVPT, where the perturbative correction is applied to a single electronic state.Research implementations has been also developed for Quasi-Degenerate cases, where a set of electronic states undergo the perturbative correction at the same time, allowing interaction among themselves. The theory development makes use of the quasi-degenerate formalism by Lindgren and the Hamiltonian multipartitioning technique from Zaitsevskii and Malrieu.
Let
(0) | |
\Psi | |
m |
(0) | |
\Psi | |
m |
=\sumI
obtained diagonalizing the true Hamiltonian
\hat{l{H}}
\hat{l{P}}\rm\hat{l{H}}\hat{l{P}}\rm
(0) | |
\left|\Psi | |
m |
\right\rangle=
(0) | |
E | |
m |
(0) | |
\left|\Psi | |
m |
\right\rangle
where
\hat{l{P}}\rm
k
-2\lek\le2
\Phic
v | |
\Psi | |
m |
(0) | |
\left|\Psi | |
m |
\right\rangle=\left|\Phic
v\right\rangle | |
\Psi | |
m |
then the perturber wavefunctions can be written as
k | |
\left|\Psi | |
l,\mu |
\right\rangle=
-k | |
\left|\Phi | |
l |
v+k | |
\Psi | |
\mu |
\right\rangle
The pattern of inactive orbitals involved in the procedure can be grouped as a collective index
l
k | |
\Psi | |
l,\mu |
\mu
Supposing indexes
i
j
a
b
r
s
k=0
k=+1
k=-1
k=+2
k=-2
These cases always represent situations where interclass electronic excitations happen. Other three excitation schemes involve a single interclass excitation plus an intraclass excitation internal to the active space:
k=0
k=+1
k=-1
A possible approach is to define the perturber wavefunctions into Hilbert spaces
k | |
S | |
l |
-k | |
\Phi | |
l |
k | |
\Psi | |
I |
k | |
S | |
l |
\stackrel{def
The full dimensionality of these spaces can be exploited to obtain the definition of the perturbers, by diagonalizing the Hamiltonian inside them
\hat{l{P}} | |||||||
|
\hat{l{H}}\hat{l{P}} | |||||||
|
-k | |
\left|\Phi | |
l |
v+k | |
\Psi | |
\mu |
\right\rangle=El,\mu
-k | |
\left|\Phi | |
l |
v+k | |
\Psi | |
\mu |
\right\rangle
This procedure is impractical given its high computational cost: for each
k | |
S | |
l |
\hat{l{H}}D
\hat{l{H}}D
-k | |
\left|\Phi | |
l |
v+k | |
\Psi | |
\mu |
\right\rangle=
k | |
E | |
l,\mu |
-k | |
\left|\Phi | |
l |
v+k | |
\Psi | |
\mu |
\right\rangle
stripping out the constant contribution of the inactive part and leaving asubsystem to be solved for the valence part
D | |
\hat{l{H}} | |
v |
v+k | |
\left|\Psi | |
\mu |
\right\rangle=
k | |
E | |
\mu |
v+k | |
\left|\Psi | |
\mu |
\right\rangle
The total energy
k | |
E | |
l,\mu |
k | |
E | |
\mu |
-k | |
\Phi | |
l |
A different choice in the development of the NEVPT approach is to choose a single function for each space
k | |
S | |
l |
\hat{l{P}} | |||||||
|
k | |
\Psi | |
l |
=
\hat{l{P}} | |||||||
|
(0) | |
\hat{l{H}}\Psi | |
m |
where
\hat{l{P}} | |||||||
|
k | |
\Psi | |
l |
=
k | |
V | |
l |
(0) | |
\Psi | |
m |
For each space, appropriate operators can be devised. We will not present their definition, as it could result overkilling. Suffice to say that the resulting perturbers are not normalized, and their norm
k | |
N | |
l |
=
k\left.\right| | |
\left\langle\Psi | |
l |
k\right\rangle | |
\Psi | |
l |
=
(0) | |
\left\langle\Psi | |
m |
\left|
k\right) | |
\left(V | |
l |
+
k | |
V | |
l |
\right|
(0) | |
\Psi | |
m |
\right\rangle
plays an important role in the Strongly Contracted development. To evaluate these norms, the spinless density matrix of rank not higher than three between the
(0) | |
\Psi | |
m |
An important property of the
k | |
\Psi | |
l |
k | |
S | |
l |
k | |
\Psi | |
l |
k | |
\Psi | |
l |
\hat{l{H}}0=\sumlk\left|
k | |
\Psi | |
l |
{}\prime\right\rangle
k | |
E | |
l |
\left\langle
k | |
\Psi | |
l |
{}\prime\right\rangle+\summ\left|
(0) | |
\Psi | |
m |
\right\rangle
(0) | |
E | |
m |
\left\langle
(0) | |
\Psi | |
m |
\right|
where
\left|
k | |
\Psi | |
l |
{}\prime\right\rangle
\left|
k | |
\Psi | |
l |
\right\rangle
The expression for the first-order correction to the wavefunction is therefore
(1) | |
\Psi | |
m |
=\sumkl\left|
k | |
\Psi | |
l |
{}\prime\right\rangle
| |||||||||
\prime |
\left|\hat{l{H}}\right|
(0) | |
\Psi | |
m |
(0) | |
\right\rangle} {E | |
m |
-
k | |
E | |
l |
and for the energy is
(2) | |
E | |
m |
=\sumkl
| |||||||||
\prime |
\left|\hat{l{H}}\right|
(0) | |
\Psi | |
m |
\right\rangle
(0) | |
\right| | |
m |
-
k | |
E | |
l |
This result still misses a definition of the perturber energies
k | |
E | |
l |
k | |
E | |
l |
=
1 | ||||||
|
\left\langle
k | |
\Psi | |
l |
\left|\hat{l{H}}D\right|
k | |
\Psi | |
l |
\right\rangle
leading to
k | |
N | |
l |
k | |
E | |
l |
=\left\langle
(0) | |
\Psi | |
m |
\left|\left(
k | |
V | |
l |
\right)+
k | |
\hat{l{H}} | |
l |
\right|
(0) | |
\Psi | |
m |
\right\rangle=\left\langle
(0) | |
\Psi | |
m |
\left|
k | |
\left(V | |
l |
\right)+
k | |
V | |
l |
\hat{l{H}}D\right|
(0) | |
\Psi | |
m |
\right\rangle+\left\langle
(0) | |
\Psi | |
m |
\left|\left(
k | |
V | |
l |
\right)+\left[\hat{l{H}}D,
k | |
V | |
l |
\right]\right|
(0) | |
\Psi | |
m |
\right\rangle
Developing the first term and extracting the inactive part of the Dyall's Hamiltonian it can be obtained
k | |
E | |
l |
=
(0) | |
E | |
m |
+\Delta\epsilonl+
1 | ||||||
|
\left\langle
(0) | |
\Psi | |
m |
\left|
k | |
\left(V | |
l |
\right)+\left[\hat{l{H}}v,
k | |
V | |
l |
\right]\right|
(0) | |
\Psi | |
m |
\right\rangle
with
\Delta\epsilonl
The term that still needs to be evaluated is the bracket involving the commutator. This can be obtained developing each
V
(0) | |
V | |
ijrs |
(2) | |
E | |
m |
\left(
0 | |
S | |
rsij |
\right)=-
| |||||||
\epsilonr+\epsilons-\epsiloni-\epsilonj |
NEVPT2 can therefore be seen as a generalized form of MP2 to multireference wavefunctions.
An alternative approach, named Partially Contracted (PC) is to define the perturber wavefunctions in a subspace
k | |
\overline{S} | |
l |
k | |
S | |
l |
\Phi
k | |
V | |
l |
-1 | |
V | |
rsi |
-1 | |
V | |
rsi |
=\gammars\suma\left(\left\langlers\left.\right|ia\right\rangleEriEsa+\left\langlesr\left.\right|ia\right\rangleEsiEra\right) r\les
The Partially Contracted approach makes use of functions
\Phirisa=EriEsa
(0) | |
\Psi | |
m |
\Phirisa=EsiEra
(0) | |
\Psi | |
m |
-1 | |
\overline{S} | |
rsi |
Once all the
k | |
\overline{S} | |
l |
\hat{l{P}}\overline{S
k}\hat{l{H}}\hat{l{P}} | |
\overline{S |
k} | |
l |
\left|
k | |
\Psi | |
l\mu |
\right\rangle
k | |
= E | |
l,\mu |
\left|
k | |
\Psi | |
l\mu |
\right\rangle
As usual, the evaluation of the Partially Contracted perturbative correction by means of the Dyall Hamiltonian involves simply manageable entities for nowadays computers.
Although the Strongly Contracted approach makes use of a perturbative space with very low flexibility, in general it provides values in very good agreement with those obtained by the more decontracted space defined for the Partially Contracted approach. This can be probably explained by the fact that the Strongly Contracted perturbers are a good average of the totally decontracted perturbative space.
The Partially Contracted evaluation has a very little overhead in computational cost with respect to the Strongly Contracted one, therefore they are normally evaluated together.
NEVPT is blessed with many important properties, making the approach very solid and reliable. These properties arise both from the theoretical approach used and on the Dyall's Hamiltonian particular structure:
NEVPT is size consistent (strict separable). Briefly, if A and B are two non-interacting systems, the energy of the supersystem A-B is equal to the sum of the energy of A plus the energy of B taken by themselves (
E(A-B)=E(A)+E(B)