Theor Chem Acc (2015) 134:100
1 3
canonical procedure [ 9 , 10 ]. Biorthogonal treatment of the
overlap when correcting a single, multideterminantal reference has been advocated and investigated extensively in the
laboratory of Péter Surján [ 11 – 13 ]. The framework termed
multiconfi gurational PT (MCPT) collects several approaches
differing in the treatment of overlap and choice for the zeroorder Hamiltonian (see Ref. [ 14 ] for an elaboration on the
perturbative partitioning). Focusing on the handling of overlap, options exploited so far, given a single reference vector
are
1. orthogonal projection to the reference (fi rst step of a
Gram–Schmidt procedure)
(a) biorthonormal set construction in the (N − 1)
-dimensional space
(b) Löwdin orthogonalization in the (N − 1) -dimensional space
2. omit orthogonal projection to the reference
(a) biorthonormal set construction in the N -dimensional space
A notable feature of all three strategies is that the
(bi)orthonormal set can be given explicitly, eliminating the
need of numerical overlap treatment. Option 2b is lucidly
missing from the above list just for the reason that no close
form of the underlying inverse square root of the overlap
could be constructed. Notation 2a is hence somewhat superfl uous, which we however keep to stress analogy with 1a.
The procedure of option 1b is equivalent to the Jacobi rotations’ inspired orthogonalization, designed by Mayer [ 15 ,
16 ]. For applications of the above (bi)orthogonal schemes in
PT strategies alternative to MCPT, see Refs. [ 17 , 18 ].
In the present work, we extend the above overlap treatments for the case of multiple reference vectors. This involves
derivation of explicit formulae for the (bi)orthonormal sets in
cases analogous to 1a, 1b, and 2a above, with reference vectors, m > 1 in number. Construction of S −1 and S −1/2 in case
1 for the (N − m) × (N − m) overlap matrix and S −1 in case
2 for the N × N overlap matrix is presented in Sect. 2 . These
results facilitate a multistate extension of the MCPT framework. The pertinent formulae are given in Sect. 3 , followed
by an illustrative numerical study in Sect. 4 .
2 (Bi)orthogonal vector sets
Let {e i }
N
i=1 be the set of unit vectors, and let us replace the
fi rst m vectors with the orthonormal set
1 of vectors {c i } m
i=1 .
1 Orthonormality of vectors c i is assumed since orthonormalizing m
vectors is relatively cheap for m N .
The new N -dimensional set, {c i } m
i=1 ∪ {e i }
N
i=m+1 is not
orthogonal as its subsets {c i } m
i=1 and {e i }
N
i=m+1 overlap.
Let us assume that coeffi cients C ji of the expansion
are arranged in matrix C , of dimension N × m . We now
introduce notation C 1 for the upper m × m block of C , and
C 2 for the lower (N − m) × m block. This allows to write
Orthonormality of vectors c i is refl ected by
where I m denotes the m -dimensional unit matrix, and shorthand A is introduced for subsequent use. We note here that the
union of sets {c i } m
i=1 and {e i }
N
i=m+1 being N -dimensional relies
on the tacit assumption that matrix A is positive defi nite.
To extend option 1, projector P corresponding to the
vector set {c i } m
i=1 is formulated as
As the next step, vectors {e i }
N
i=m+1 , arranged as the last
N − m columns of unit matrix I N , are projected orthogonal
to {c i } m
i=1 to generate the set {e i }
N
i=m+1 . Denoting the corresponding matrix D , we get
with obvious notation for I N−m . Overlap of vectors e i can
be expressed as
using Eqs. ( 4 ) and ( 5 ). Matrix D † D represents the nontrivial part of the N × N overlap matrix
of the set {c i } m
i=1 ∪ {e i }
N
i=m+1 .
In the following paragraphs, the inverse and inverse
square root of the overlap matrix of Eq. ( 7 ) is constructed.
Clearly, it suffi ces to focus on D † D of Eq. ( 6 ). We
approach the problem in a general manner, by expressing
any analytic function, f : (0, 1] → R of matrix S . Matrix
function f (S) is defi ned via Taylor expansion, written as
c
i
=
N
j=1
e
j C ji
(3)
C =
C 1
C 2
.
(4)
I m = C
†
1 C 1 + C
†
2 C 2 = A + C
†
2 C 2 ,
P = CC
†
=
C 1 C
†
1 C 1 C
†
2
C 2 C
†
1 C 2 C
†
2
.
(5)
D
= (I N − P)
0
I N−m
=
− C 1 C
†
2
I N−m − C 2 C
†
2
,
(6)
D
† D
= I N−m − C 2 C
†
2 ,
(7)
S =
I m
D † D
(8)
f
I N−m − C 2 C
†
2
=
∞
n=0
d n
C 2 C
†
2
n
,
230
Reprinted from the journal
1 3
canonical procedure [ 9 , 10 ]. Biorthogonal treatment of the
overlap when correcting a single, multideterminantal reference has been advocated and investigated extensively in the
laboratory of Péter Surján [ 11 – 13 ]. The framework termed
multiconfi gurational PT (MCPT) collects several approaches
differing in the treatment of overlap and choice for the zeroorder Hamiltonian (see Ref. [ 14 ] for an elaboration on the
perturbative partitioning). Focusing on the handling of overlap, options exploited so far, given a single reference vector
are
1. orthogonal projection to the reference (fi rst step of a
Gram–Schmidt procedure)
(a) biorthonormal set construction in the (N − 1)
-dimensional space
(b) Löwdin orthogonalization in the (N − 1) -dimensional space
2. omit orthogonal projection to the reference
(a) biorthonormal set construction in the N -dimensional space
A notable feature of all three strategies is that the
(bi)orthonormal set can be given explicitly, eliminating the
need of numerical overlap treatment. Option 2b is lucidly
missing from the above list just for the reason that no close
form of the underlying inverse square root of the overlap
could be constructed. Notation 2a is hence somewhat superfl uous, which we however keep to stress analogy with 1a.
The procedure of option 1b is equivalent to the Jacobi rotations’ inspired orthogonalization, designed by Mayer [ 15 ,
16 ]. For applications of the above (bi)orthogonal schemes in
PT strategies alternative to MCPT, see Refs. [ 17 , 18 ].
In the present work, we extend the above overlap treatments for the case of multiple reference vectors. This involves
derivation of explicit formulae for the (bi)orthonormal sets in
cases analogous to 1a, 1b, and 2a above, with reference vectors, m > 1 in number. Construction of S −1 and S −1/2 in case
1 for the (N − m) × (N − m) overlap matrix and S −1 in case
2 for the N × N overlap matrix is presented in Sect. 2 . These
results facilitate a multistate extension of the MCPT framework. The pertinent formulae are given in Sect. 3 , followed
by an illustrative numerical study in Sect. 4 .
2 (Bi)orthogonal vector sets
Let {e i }
N
i=1 be the set of unit vectors, and let us replace the
fi rst m vectors with the orthonormal set
1 of vectors {c i } m
i=1 .
1 Orthonormality of vectors c i is assumed since orthonormalizing m
vectors is relatively cheap for m N .
The new N -dimensional set, {c i } m
i=1 ∪ {e i }
N
i=m+1 is not
orthogonal as its subsets {c i } m
i=1 and {e i }
N
i=m+1 overlap.
Let us assume that coeffi cients C ji of the expansion
are arranged in matrix C , of dimension N × m . We now
introduce notation C 1 for the upper m × m block of C , and
C 2 for the lower (N − m) × m block. This allows to write
Orthonormality of vectors c i is refl ected by
where I m denotes the m -dimensional unit matrix, and shorthand A is introduced for subsequent use. We note here that the
union of sets {c i } m
i=1 and {e i }
N
i=m+1 being N -dimensional relies
on the tacit assumption that matrix A is positive defi nite.
To extend option 1, projector P corresponding to the
vector set {c i } m
i=1 is formulated as
As the next step, vectors {e i }
N
i=m+1 , arranged as the last
N − m columns of unit matrix I N , are projected orthogonal
to {c i } m
i=1 to generate the set {e i }
N
i=m+1 . Denoting the corresponding matrix D , we get
with obvious notation for I N−m . Overlap of vectors e i can
be expressed as
using Eqs. ( 4 ) and ( 5 ). Matrix D † D represents the nontrivial part of the N × N overlap matrix
of the set {c i } m
i=1 ∪ {e i }
N
i=m+1 .
In the following paragraphs, the inverse and inverse
square root of the overlap matrix of Eq. ( 7 ) is constructed.
Clearly, it suffi ces to focus on D † D of Eq. ( 6 ). We
approach the problem in a general manner, by expressing
any analytic function, f : (0, 1] → R of matrix S . Matrix
function f (S) is defi ned via Taylor expansion, written as
c
i
=
N
j=1
e
j C ji
(3)
C =
C 1
C 2
.
(4)
I m = C
†
1 C 1 + C
†
2 C 2 = A + C
†
2 C 2 ,
P = CC
†
=
C 1 C
†
1 C 1 C
†
2
C 2 C
†
1 C 2 C
†
2
.
(5)
D
= (I N − P)
0
I N−m
=
− C 1 C
†
2
I N−m − C 2 C
†
2
,
(6)
D
† D
= I N−m − C 2 C
†
2 ,
(7)
S =
I m
D † D
(8)
f
I N−m − C 2 C
†
2
=
∞
n=0
d n
C 2 C
†
2
n
,
230
Reprinted from the journal