Theor Chem Acc (2015) 134:134
1 3
respectively, where the ordering of the nuclei as components of these vectors is consistent with the common
nuclear geometry r .
If z is a convex combination of the other n nuclear
charge vectors, z
(1)
, z
(2)
,… z
( i )
,… z
( n ) , that is, if
where
and where
then due to linearity of the electronic Hamiltonians H e ( z ,
r ) in components of z , a relation analogous to Eq. ( 5 ) must
apply for their corresponding electronic Hamiltonians as
well,
If the electronic wavefunction of system M is Ψ e , then
based on the above, the electronic energy expectation value
of system M can be written as
However, the wavefunction Ψ e of system M is not in general variationally optimal for any of the other molecular
systems M
(1) , M
(2)
,… M
( i ) ,… M
( n )
. Consequently, according to the variational theorem, the right-hand side of Eq. ( 9 )
cannot increase if in the expectation value expression for
each of the right-hand side Hamiltonians H e ( z
( i ) , r ), we
replace Ψ e with the actual wavefunction Ψ e
( i ) of the respective molecular system M
( i )
.
Consequently, we obtain that
But, inequality ( 10 ) is in fact an inequality for the electronic energies,
That is, convexity for the set of nuclear charge vectors,
which in our case form a simplex in an ( n − 1)-dimensional nuclear charge space Z , implies convexity for the
respective electronic energies.
Specifi cally, if the nuclear charge vector z falls on or
inside of the Z -space simplex defi ned by the nuclear charge
(5)
z = α
(1) z
(1)
+ α
(2) z
(2)
+ · · · + α
(i) z
(i)
+ · · · + α
(n) z
(n) ,
(6)
0 ≤ α
(i)
≤ 1, for every i,
(7)
α
(1)
+ α
(2)
+ · · · + α
(i)
+ · · · + α
(n)
= 1
(8)
H e (z, r) = α
(1) H e
z
(1) , r
+ α
(2) H e
z
(2) , r
+ · · · + α
(i) H e
z
(i) , r
+ · · · + α
(n) H e
z
(n) , r
,
(9)
Ψ e |H e (z, r)|Ψ e =
i=1,n
α
(i)
Ψ e |H e
z
(i) , r
|Ψ e
(10)
Ψ e |H e (z, r)|Ψ e ≥
i=1,n
α
(i)
Ψ
(i)
e |H e
z
(i) , r
|Ψ
(i)
e
(11)
E e ≥
i=1,n
α
(i) E
(i)
e
vectors z
(1)
, z
(2)
,… z
( i )
,… z
( n )
, then the electronic energy
relation ( 11 ) must hold.
The special case of n = 2 has been studied in most
detail, and one of the simplest applications that has been
already discussed is the comparison of the electronic
energies of N 2 , CO, and the equivalent OC molecule. If
M = N 2 , M
(1)
= CO, and M
(2)
= OC, with identical bond
lengths, and with the following 2D nuclear charge vectors
z = (7, 7)′, z
(1)
= (6, 8)′ and z
(2)
= (8, 6)′, respectively, then
these nuclear charge vectors fulfi ll the convexity condition
where, as in general, 0 ≤ α ≤ 1, now with the simple special choice of α = 0.5.
Consequently, the theorem applies, hence for every common bond length, that is, for the entire electronic potential
energy curves
that is,
E e (N 2 ) ≥ E e (CO) for every common bond length.
Note that this is a rigorous (although simple) quantum
chemical result, as derived entirely by using nuclear charge
variations.
However, convexity, if used in a judicious way, can also
be used for extrapolation!
Consider now the case of M = CO, M
(1)
= N 2 , and
M
(2)
= BF triple of diatomics, again, with identical bond
lengths, and with the following 2D nuclear charge vectors
z = (6, 8), z
(1)
= (7, 7), and z
(2)
= (5, 9), respectively.
Again, we obtain that
where 0 ≤ α ≤ 1, with the current special choice of α = 0.5.
Hence, just as before, the theorem applies, consequently,
for every common bond length, that is, for the entire electronic potential energy curves
However, now we can combine this result with the previous one,
and we get
that implies
z = αz
(1)
+ (1−α) z
(2)
E e ≥ 0.5E
(1)
e + (1 − 0.5)E
(2)
e
E e (N 2 ) ≥ 0.5E
(1)
e (CO) + (1 − 0.5)E
(2)
e (OC),
z = αz
(1)
+ (1−α) z
(2)
E e ≥ 0.5E
(1)
e + 0.5E
(2)
e
E e (CO) ≥ 0.5E
(1)
e (N 2 ) + 0.5E
(2)
e (BF).
E e (N 2 ) ≥ E e (CO),
E e (N 2 ) ≥ E e (CO) ≥ 0.5E
(1)
e (N 2 ) + 0.5E
(2)
e (BF),
27
Reprinted from the journal
1 3
respectively, where the ordering of the nuclei as components of these vectors is consistent with the common
nuclear geometry r .
If z is a convex combination of the other n nuclear
charge vectors, z
(1)
, z
(2)
,… z
( i )
,… z
( n ) , that is, if
where
and where
then due to linearity of the electronic Hamiltonians H e ( z ,
r ) in components of z , a relation analogous to Eq. ( 5 ) must
apply for their corresponding electronic Hamiltonians as
well,
If the electronic wavefunction of system M is Ψ e , then
based on the above, the electronic energy expectation value
of system M can be written as
However, the wavefunction Ψ e of system M is not in general variationally optimal for any of the other molecular
systems M
(1) , M
(2)
,… M
( i ) ,… M
( n )
. Consequently, according to the variational theorem, the right-hand side of Eq. ( 9 )
cannot increase if in the expectation value expression for
each of the right-hand side Hamiltonians H e ( z
( i ) , r ), we
replace Ψ e with the actual wavefunction Ψ e
( i ) of the respective molecular system M
( i )
.
Consequently, we obtain that
But, inequality ( 10 ) is in fact an inequality for the electronic energies,
That is, convexity for the set of nuclear charge vectors,
which in our case form a simplex in an ( n − 1)-dimensional nuclear charge space Z , implies convexity for the
respective electronic energies.
Specifi cally, if the nuclear charge vector z falls on or
inside of the Z -space simplex defi ned by the nuclear charge
(5)
z = α
(1) z
(1)
+ α
(2) z
(2)
+ · · · + α
(i) z
(i)
+ · · · + α
(n) z
(n) ,
(6)
0 ≤ α
(i)
≤ 1, for every i,
(7)
α
(1)
+ α
(2)
+ · · · + α
(i)
+ · · · + α
(n)
= 1
(8)
H e (z, r) = α
(1) H e
z
(1) , r
+ α
(2) H e
z
(2) , r
+ · · · + α
(i) H e
z
(i) , r
+ · · · + α
(n) H e
z
(n) , r
,
(9)
Ψ e |H e (z, r)|Ψ e =
i=1,n
α
(i)
Ψ e |H e
z
(i) , r
|Ψ e
(10)
Ψ e |H e (z, r)|Ψ e ≥
i=1,n
α
(i)
Ψ
(i)
e |H e
z
(i) , r
|Ψ
(i)
e
(11)
E e ≥
i=1,n
α
(i) E
(i)
e
vectors z
(1)
, z
(2)
,… z
( i )
,… z
( n )
, then the electronic energy
relation ( 11 ) must hold.
The special case of n = 2 has been studied in most
detail, and one of the simplest applications that has been
already discussed is the comparison of the electronic
energies of N 2 , CO, and the equivalent OC molecule. If
M = N 2 , M
(1)
= CO, and M
(2)
= OC, with identical bond
lengths, and with the following 2D nuclear charge vectors
z = (7, 7)′, z
(1)
= (6, 8)′ and z
(2)
= (8, 6)′, respectively, then
these nuclear charge vectors fulfi ll the convexity condition
where, as in general, 0 ≤ α ≤ 1, now with the simple special choice of α = 0.5.
Consequently, the theorem applies, hence for every common bond length, that is, for the entire electronic potential
energy curves
that is,
E e (N 2 ) ≥ E e (CO) for every common bond length.
Note that this is a rigorous (although simple) quantum
chemical result, as derived entirely by using nuclear charge
variations.
However, convexity, if used in a judicious way, can also
be used for extrapolation!
Consider now the case of M = CO, M
(1)
= N 2 , and
M
(2)
= BF triple of diatomics, again, with identical bond
lengths, and with the following 2D nuclear charge vectors
z = (6, 8), z
(1)
= (7, 7), and z
(2)
= (5, 9), respectively.
Again, we obtain that
where 0 ≤ α ≤ 1, with the current special choice of α = 0.5.
Hence, just as before, the theorem applies, consequently,
for every common bond length, that is, for the entire electronic potential energy curves
However, now we can combine this result with the previous one,
and we get
that implies
z = αz
(1)
+ (1−α) z
(2)
E e ≥ 0.5E
(1)
e + (1 − 0.5)E
(2)
e
E e (N 2 ) ≥ 0.5E
(1)
e (CO) + (1 − 0.5)E
(2)
e (OC),
z = αz
(1)
+ (1−α) z
(2)
E e ≥ 0.5E
(1)
e + 0.5E
(2)
e
E e (CO) ≥ 0.5E
(1)
e (N 2 ) + 0.5E
(2)
e (BF).
E e (N 2 ) ≥ E e (CO),
E e (N 2 ) ≥ E e (CO) ≥ 0.5E
(1)
e (N 2 ) + 0.5E
(2)
e (BF),
27
Reprinted from the journal