where the first term refers to the quantum-mechanically uncorrelated Coulomb-like
pair density, the second term to the Fock-Dirac exchange (which is dominated by
and associated with the Fermi hole), while the third term is at least an order of
magnitude smaller [41, 68] than the second term, and connected with the Coulomb
hole. The energy quantity V
AB
ee is associated with the whole of ρ 2 (r 1 ,r 2 ), collecting
the three types of interactions that electrons experience when interacting with each
other. Each term in Eq. 2.16 is associated with a type of potential energy, so that the
corresponding fine-structure of V
AB
ee automatically follows,
V
AB
ee ¼
Z
XA
dr 1
Z
XB
dr 2
q 2 ðr 1 ; r 2 Þ
r 12
¼
Z
XA
dr 1
Z
XB
dr 2
qðr 1 Þqðr 2 Þ
r 12
À
Z
XA
dr 1
Z
XB
dr 2
q 1 ðr 1 ; r 2 Þq 1 ðr 2 ; r 1 Þ
r 12
þ
Z
XA
dr 1
Z
XB
dr 2
q
corr
2 ðr 1 ; r 2 Þ
r 12
¼ V
AB
ee;coul þ V
AB
ee;exch þ V
AB
ee;corr
ð2:17Þ
The second term in Eq. 2.17 represents the exchange delocalisation energy,
V
AB
ee;exch , which is (already) present at Hartree-Fock level. This term teases out the
interaction that keeps bonded atoms together. The degree to which atoms are bonded
can be estimated by a non-energy measure, which is typically a quantum-mechanical
bond order. QCT offers such a measure [69]. However, it was shown by our lab [56]
that this bond order is only the first term of the multipolar expansion of V
AB
ee;exch .
Hence, the latter quantity contains more information than a bond order. However, in
the construction of QCTFF, the route of expanding V
AB
ee;exch as so-called exchange
moments was abandoned because they have an imprint of the molecular orbitals they
are derived from. This imprint hampers transferability. The energy quantity V
AB
ee;exch
can remain unexpanded because it drops off so quickly with distance [70] in saturated systems, which proteins largely are. However, multipole moments are essential
in the representation of electrostatics because this type of interaction drops off more
slowly than V
AB
ee;exch . Therefore the number of non-negligible V
AB
elec values is much
larger than the number of V
AB
ee;exch values. The trouble with this observation is the
rapidly increase in the number of possible distances between A and B. In other
words, atoms that are further apart can appear in more possible configurations than
atoms that are closer to each other. This is why it is undesirable to calculate all
possible 1,n (n > 4) V
AB
elec interactions. A multipole series succeeds in avoiding the
calculation of all these V
AB
elec interactions. The series separates a geometrically
entangled (since r 12 involves simultaneously r 1 and r 2 ) energy quantity into single
atom quantities, i.e. multipole moments. This separation enables the calculation of
the interatomic interaction to be free of large geometric variations. Conversely,
short-range interactions (1, 2; 1, 3 and 1, 4) are geometrically much more constrained and hence would not benefit that much from multipole moments. This is
why it is alright to not expand the electrostatic energy V
AB
elec at short range.
40
P.L.A. Popelier
pair density, the second term to the Fock-Dirac exchange (which is dominated by
and associated with the Fermi hole), while the third term is at least an order of
magnitude smaller [41, 68] than the second term, and connected with the Coulomb
hole. The energy quantity V
AB
ee is associated with the whole of ρ 2 (r 1 ,r 2 ), collecting
the three types of interactions that electrons experience when interacting with each
other. Each term in Eq. 2.16 is associated with a type of potential energy, so that the
corresponding fine-structure of V
AB
ee automatically follows,
V
AB
ee ¼
Z
XA
dr 1
Z
XB
dr 2
q 2 ðr 1 ; r 2 Þ
r 12
¼
Z
XA
dr 1
Z
XB
dr 2
qðr 1 Þqðr 2 Þ
r 12
À
Z
XA
dr 1
Z
XB
dr 2
q 1 ðr 1 ; r 2 Þq 1 ðr 2 ; r 1 Þ
r 12
þ
Z
XA
dr 1
Z
XB
dr 2
q
corr
2 ðr 1 ; r 2 Þ
r 12
¼ V
AB
ee;coul þ V
AB
ee;exch þ V
AB
ee;corr
ð2:17Þ
The second term in Eq. 2.17 represents the exchange delocalisation energy,
V
AB
ee;exch , which is (already) present at Hartree-Fock level. This term teases out the
interaction that keeps bonded atoms together. The degree to which atoms are bonded
can be estimated by a non-energy measure, which is typically a quantum-mechanical
bond order. QCT offers such a measure [69]. However, it was shown by our lab [56]
that this bond order is only the first term of the multipolar expansion of V
AB
ee;exch .
Hence, the latter quantity contains more information than a bond order. However, in
the construction of QCTFF, the route of expanding V
AB
ee;exch as so-called exchange
moments was abandoned because they have an imprint of the molecular orbitals they
are derived from. This imprint hampers transferability. The energy quantity V
AB
ee;exch
can remain unexpanded because it drops off so quickly with distance [70] in saturated systems, which proteins largely are. However, multipole moments are essential
in the representation of electrostatics because this type of interaction drops off more
slowly than V
AB
ee;exch . Therefore the number of non-negligible V
AB
elec values is much
larger than the number of V
AB
ee;exch values. The trouble with this observation is the
rapidly increase in the number of possible distances between A and B. In other
words, atoms that are further apart can appear in more possible configurations than
atoms that are closer to each other. This is why it is undesirable to calculate all
possible 1,n (n > 4) V
AB
elec interactions. A multipole series succeeds in avoiding the
calculation of all these V
AB
elec interactions. The series separates a geometrically
entangled (since r 12 involves simultaneously r 1 and r 2 ) energy quantity into single
atom quantities, i.e. multipole moments. This separation enables the calculation of
the interatomic interaction to be free of large geometric variations. Conversely,
short-range interactions (1, 2; 1, 3 and 1, 4) are geometrically much more constrained and hence would not benefit that much from multipole moments. This is
why it is alright to not expand the electrostatic energy V
AB
elec at short range.
40
P.L.A. Popelier