234
Appendix E: Solutions
a ferromagnetic interaction between the two units. fully aligned: All carbon atoms
form shortest contacts and the product of atomic spin populations is positive in all
cases, hence an antiferromagnetic coupling can be expected.
Exercise 4.4 The symmetry of the complex is D 2h and the five d-orbitals by increasing orbital energy are 3d xz (b 2g ), 3d yz (b 3g ), 3d xy (b 1g ), 3d z 2 (a g ), 3d x 2 −y 2 (a g ), with
the irreducible representation in parentheses. Cr III has a d 3 electronic configuration, occupying the b 1g , b 2g and b 3g orbitals. Ni II has a d 8 electronic configuration
and the orbitals with unpaired electrons are the two a g ’s. All exchange paths that
connect Cr with Ni involve orbitals of different symmetries, implying zero overlap
between the magnetic orbitals, and hence, ferromagnetic coupling. Replacing Cr III
with Mn II introduces two extra unpaired electrons on site A, and hence, 5 × 2 = 10
exchange paths. Among the ten exchange paths, the four involving the a g orbitals will
give a (strong) antiferromagnetic contribution, which counterbalances the (weaker)
ferromagnetic contribution of the other six paths. The net coupling will be antiferromagnetic.
Exercise 4.5 There are six determinants with two doubly occupied orbitals, six
M S = 0 determinants can be constructed with all orbitals singly occupied and the
distribution with one doubly occupied orbital, two singly occupied and one empty
orbital can be realized in 24 different ways. The total CAS wave function is a linear
combination of 36 different determinants. Note that the use of spin symmetry reduces
the expansion to 20 CSFs for S = 0, 15 CSFs for S = 1 and 1 CSF for S = 2.
Exercise 4.6 We have four reference determinants |...aa|, |...ba|, |...ab| and
|...ba|. There are 2k occupied spin orbitals which can be replaced by one of the 2l
unoccupied spinorbitals: 2k × l × 4. In addition there are also replacements involving
a spin-flip, i.e. α → β and β → α. This has to be compensated by spin-flip in the
CAS space, which can only be done for |...ab| and |...ba|:2 k × l × 2. In total:
12 × k × l.
Exercise 4.7 The CAS(2, 2) reference contains 4 M S = 0 determinants; n = 4.
There are 72 electrons in the inactive orbitals, k = 36. The system has 154 − 36 − 2
virtual orbitals, l = 116. This results in 36 2 ×116×4 = 601344 2h-1p determinants,
36 × 116 2 × 4 = 1937664 1h-2p determinants, and 36 2 × 116 2 × 4 = 69755904 2h2p determinants. The total number of determinants in the MR-CISD wave function
is ≈72294912, from which the 2h-2p determinants constitute more than 96 %.
Exercise 4.8 Φ Q =| ppbb|. The contribution to Φ I | ˆ
H eff |Φ L equals:
hhab| ˆ
V |ppbbppbb| ˆ
V |hhbb
E L −E Q
. The first matrix element is zero since the determinants Φ I
and Φ Q differ by more than two columns.
Exercise 4.9 (a) Neutral determinant Φ I =| hhab|; ionic determinant Φ J =
|hhaa|. Because both Φ I and Φ J have two different columns with respect to Φ R ,
Φ I | ˆ
H|Φ R Φ R | ˆ
H|Φ J is non-zero and in consequence the effective matrix element
between Φ I and Φ J will be different from zero (see Eq. 1.86). (b) Since the matrix
Appendix E: Solutions
a ferromagnetic interaction between the two units. fully aligned: All carbon atoms
form shortest contacts and the product of atomic spin populations is positive in all
cases, hence an antiferromagnetic coupling can be expected.
Exercise 4.4 The symmetry of the complex is D 2h and the five d-orbitals by increasing orbital energy are 3d xz (b 2g ), 3d yz (b 3g ), 3d xy (b 1g ), 3d z 2 (a g ), 3d x 2 −y 2 (a g ), with
the irreducible representation in parentheses. Cr III has a d 3 electronic configuration, occupying the b 1g , b 2g and b 3g orbitals. Ni II has a d 8 electronic configuration
and the orbitals with unpaired electrons are the two a g ’s. All exchange paths that
connect Cr with Ni involve orbitals of different symmetries, implying zero overlap
between the magnetic orbitals, and hence, ferromagnetic coupling. Replacing Cr III
with Mn II introduces two extra unpaired electrons on site A, and hence, 5 × 2 = 10
exchange paths. Among the ten exchange paths, the four involving the a g orbitals will
give a (strong) antiferromagnetic contribution, which counterbalances the (weaker)
ferromagnetic contribution of the other six paths. The net coupling will be antiferromagnetic.
Exercise 4.5 There are six determinants with two doubly occupied orbitals, six
M S = 0 determinants can be constructed with all orbitals singly occupied and the
distribution with one doubly occupied orbital, two singly occupied and one empty
orbital can be realized in 24 different ways. The total CAS wave function is a linear
combination of 36 different determinants. Note that the use of spin symmetry reduces
the expansion to 20 CSFs for S = 0, 15 CSFs for S = 1 and 1 CSF for S = 2.
Exercise 4.6 We have four reference determinants |...aa|, |...ba|, |...ab| and
|...ba|. There are 2k occupied spin orbitals which can be replaced by one of the 2l
unoccupied spinorbitals: 2k × l × 4. In addition there are also replacements involving
a spin-flip, i.e. α → β and β → α. This has to be compensated by spin-flip in the
CAS space, which can only be done for |...ab| and |...ba|:2 k × l × 2. In total:
12 × k × l.
Exercise 4.7 The CAS(2, 2) reference contains 4 M S = 0 determinants; n = 4.
There are 72 electrons in the inactive orbitals, k = 36. The system has 154 − 36 − 2
virtual orbitals, l = 116. This results in 36 2 ×116×4 = 601344 2h-1p determinants,
36 × 116 2 × 4 = 1937664 1h-2p determinants, and 36 2 × 116 2 × 4 = 69755904 2h2p determinants. The total number of determinants in the MR-CISD wave function
is ≈72294912, from which the 2h-2p determinants constitute more than 96 %.
Exercise 4.8 Φ Q =| ppbb|. The contribution to Φ I | ˆ
H eff |Φ L equals:
hhab| ˆ
V |ppbbppbb| ˆ
V |hhbb
E L −E Q
. The first matrix element is zero since the determinants Φ I
and Φ Q differ by more than two columns.
Exercise 4.9 (a) Neutral determinant Φ I =| hhab|; ionic determinant Φ J =
|hhaa|. Because both Φ I and Φ J have two different columns with respect to Φ R ,
Φ I | ˆ
H|Φ R Φ R | ˆ
H|Φ J is non-zero and in consequence the effective matrix element
between Φ I and Φ J will be different from zero (see Eq. 1.86). (b) Since the matrix