6 Multi -- Procrustes Fragmentation with Trigonometric Weighting
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154
Z. Szekeres and P.G. Mezey
The edge length of each cube is taken as π. The centers of cubes for the eight
different tilings are given by the vectors
r u,v,w (i,j,k) = [(i + 0.5u)π, (j + 0.5v)π, (k + 0.5w)π],
(7-13)
where these A, B, C, D, E, F, G, and H tilings follow the lexicographic order of
triples
(u, v, w), with the integers u, v, and w fulfilling the condition
0 ≤ u, v, w ≤ 1,
(7-14)
for example, the integer triple u = 1, v = 0 and w = 1 corresponds to tiling scheme
F, and where indices i, j, and k refer to the serial numbers of the locations of the
centers of the cubes along the x, y, and z coordinate axes, respectively.
The x, y, and z coordinates of the centers of all the cubes in the eight Procrustes
tiling schemes are given below:
A
B
C
D
E
F
G
H
x iπ
iπ
iπ
iπ
(i + 0.5)π (i + 0.5)π (i + 0.5)π (i + 0.5)π
y jπ
jπ
(j + 0.5)π (j + 0.5)π
jπ
jπ
(j + 0.5)π (j + 0.5)π
z kπ (k + 0.5)π
kπ
(k + 0.5)π
kπ
(k + 0.5)π
kπ
(k + 0.5)π
(7-15)
The next step is to carry out an ADMA (or MEDLA) linear scaling computation
of the macromolecule M for each of the eight tiling schemes A . . . H the usual way.
As a result, the eight direct ADMA electron densities by fragment generation
using tilings A, B, C, D, E, F, G, and H are:
ρA (x, y, z), ρB (x, y, z), ρC (x, y, z), ρD (x, y, z), ρE (x, y, z), ρF (x, y, z),
ρG (x, y, z), and ρH (x, y, z), respectively
(7-16)
Using these electron densities, a trigonometric weighting ensures, that each
boundary location of each tile of each tiling scheme contributes by a factor of zero,
whereas each more accurate center location of each tile of each tiling scheme contributes by a factor of 1, with a smooth interpolation for the weights of various
contributions for intermediate locations. The actual trigonometric formula for this
Multi-Procrustes approach is as follows:
ρMP (x,y,z) = ρA (x,y,z)cos2 xcos2 ycos2 z + ρB (x,y,z)cos2 xcos2 ysin2 z
+ρC (x,y,z)cos2 xsin2 ycos2 z + ρD (x,y,z)cos2 xsin2 ysin2 z
+ρE (x,y,z)sin2 xcos2 ycos2 z + ρF (x,y,z)sin2 xcos2 ysin2 z
+ρG (x,y,z)sin2 xsin2 ycos2 z + ρH (x,y,z)sin2 xsin2 ysin2 z
(7-17)
Although the above Multi-Procrustes approach of linear scaling macromolecular electron density generation requires an approximately eightfold increase in the
computer time needed for the same macromolecule, the weighting scheme ensures
increased accuracy. Note that, the time needed for the step involving the actual combination of electron densities according to Eq. (7-17) is negligible as compared to
Fragmentation Selection Strategies
155
the computer time needed for the actual ADMA or MEDLA computation of individual electron densities for each tiling scheme. The approach provides a possible
tool in various applications [41–43].
7.7.
SUMMARY
The role of molecular fragmentation principles in linear scaling quantum chemical
approaches, theoretical and practical conditions for their implementation, as well as
computational methods for efficient fragmentation schemes are discussed.
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CHAPTER 8
APPROXIMATIONS OF LONG-RANGE INTERACTIONS IN
FRAGMENT-BASED QUANTUM CHEMICAL APPROACHES
SIMON M. ECKARD2 , ANDREA FRANK1 , IONUT ONILA1 ,
AND THOMAS E. EXNER1
1 Department of Chemistry and Zukunftskolleg, University of Konstanz D-78457, Konstanz, Germany,
e-mail: andrea.frank@uni-konstanz.de; ionut.onila@uni-konstanz.de; thomas.exner@uni-konstanz.de
2 Institute of Physical Chemistry, University of Zurich, 8057 Zürich, Switzerland,
e-mail: Simon.Eckard@pci.uzh.ch
Abstract:
Quantum chemical calculations of very large systems still pose major challenges due
to the formidable scaling behavior of standard methods with system size. Here, we
will describe how the concept of separating short- and long-range interactions can be
used to make such calculations possible nonetheless at least in an approximate way. In
mixed quantum mechanical/molecular mechanical (QM/MM) and fragment-based quantum chemical methods, the local surroundings are considered explicitly whereas other
parts further away are neglected or included with a lower level of theory, e.g. as interactions with point charges. Different methods to combine these two descriptions, so-called
embedding schemes, are outlined. Additionally, the border region problem, how subsystems describable by quantum mechanics can be generated by cleaving and saturating
bonds connecting atoms located in the different regions, and proposed solutions are discussed. Finally, with the fragment-based adjustable density matrix assembler (ADMA)
method as example, the capacities but also some limitations of the presented approaches
will be presented using different test systems.
Keywords:
Fragment-based approaches, Distance dependence, Linear scaling, Embedding
schemes, Border regions, Adjustable density matrix assembler
8.1.
INTRODUCTION
Knowledge about the 3D structure (in atomic resolution) and the physico-chemical
properties of proteins as well as protein-protein and protein-ligand complexes is
a precondition for the in-depth understanding of biological processes and rational
manipulation of these. Because of the large size of these molecular systems, the
determination of the structure is a playground of experimental methods like X-ray
crystallography and NMR spectroscopy. But if such structures are available, theoretical methods come into play. Beside empirical methods, linear-scaling quantum
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DOI 10.1007/978-90-481-2853-2_8, C Springer Science+Business Media B.V. 2011
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chemical approaches should be considered for the analysis of such systems since
they have, as the name implies, a much better scaling behavior compared to standard
quantum chemical methods and a broader field of application without large parameterization compared to e.g. molecular-mechanical force fields. We will concentrate
here on one specific type of linear scaling methods, fragment-based quantum chemistry, and describe the approximations leading to linear scaling. Additionally, the
limitations of earlier versions as well as possible workarounds will be discussed.
8.2.
SHORT-RANGE AND LONG-RANGE INTERACTIONS
Hartree-Fock theory still plays a dominant role in quantum chemistry, due to its
relative efficiency as well as being a useful reference for methods which include
dynamic electron correlation (except density functional theory). In this theory, the
approximate equivalent of the Hamiltonian, the Fock matrix, consists of the kinetic
energy term of the electrons, the electron-nucleus interaction and the two-electron
interaction matrix G. The calculation of the latter is the most time-consuming
step since it contains the different integrals between any four of all the gaussian
basis functions used in the truncated linear combination of atomic orbitals (LCAO)
expansion series of the atomic (and molecular) orbitals, determined by the basis set
in use. These four-center two-electron integrals cause the formidable scaling behavior of O(n4 ). In the nowadays mostly applied Kohn-Sham representation of density
functional theory, equivalent considerations result in the same scaling behavior just
by replacing the Fock matrix with the Kohn-Sham matrix, even though scaling of
O(n3 ) can be obtained by density-fitting approaches [1–3].
For the case of 1s functions (the easiest case in terms of integral evaluation), the
integration of the kinetic terms Tij = ϕi | − 12 ϕj yields a distance dependency
roughly (i.e., for sufficiently large electron-electron distances rij ) proportional to
2
2
r2 · e−qr . The electron-nucleus attraction scales with 1r · e−qe · erf(r) while the
matrix elements Gijkl show a distance dependency proportional to
2
1 −arij2 −brkl
.
2 e
rijkl
The two-electron interaction shows the strongest decrease with increasing distance.
Therefore, for these an approximation by only calculating short range interactions
seems possible and is highly desirable due to the immense number of matrix elements for large molecules and/or large basis sets. Standard implementations of
Hartree-Fock already use this fact by first approximately evaluating all the integrals and then only fully calculate those which are above a specific cut-off resulting
in a scaling of roughly O(n2 · log n) [4, 5].
Since even the integration of the rather simple interaction between two 1s orbitals
results in the quite complicated derivation of the above distance dependencies, it is
not a trivial task to evaluate the behavior for combinations of functions of higher
angular momentum. Furthermore, one has to estimate the absolute values of all the
interactions for each pair of particles and then decide which of these should be fully
evaluated. Even if there are efficient methods for doing so [6–8], this becomes a too
time-consuming process to be of use for very large systems which limits the use of
pure quantum mechanics in biochemistry or material science.
Approximations of Long-Range Interactions
159
If we make the transition to perfectly localized electrons by replacing the
Gaussian basis functions by Dirac delta functions (the consequences of which
are clearly comprehensible if we let the exponents q,a,b→∞ in the derivation of
the above formula, equivalent to a view from very far away), we end up with
the classical Coulomb interaction terms for the electron-nucleus and electronelectron interaction, with the specific “roughness” of the Hamiltonian due to the
Gijkl smoothed out. This shows quite clearly that, while one needs to calculate
the exact integrals for all the interactions at short range, it suffices to consider
only the classical Coulomb interactions at long range where the 1/r behavior falls
off significantly slower than the Gaussian functions. The attributes short-ranged
and long-ranged are, of course, subject to discussion and their assignment to the
different types of interactions strongly depends on the desired level of accuracy.
Hierarchical multipole methods [9–16] generate a smooth transition between the
short-range and long-range description. They recursively subdivide a system into
a hierarchy of cells. The particle distributions within a cell are approximated by a
multipole expansion. Since this expansion become increasingly more accurate with
separations, larger cells may be used as the interaction distance increases leading
to a linear-scaling calculation of the Fock matrix when using fast multipole methods [11, 17–21]. Since this still does not solve the problem of Fock/Kohn-Sham
matrix diagonalization needed in the self-consistent cycles of HF or DFT calculations, the even stronger approximation may be applied that the electron-electron
interactions are only included up to a specific threshold and from then on they
are only implicitly included, described by Coulomb interactions with atomic point
charges, or are neglected completely. That this approximation still gives reasonable
results is demonstrated by the successful application of mixed quantum mechanical/molecular mechanical (QM/MM) and fragment-based quantum mechanical
methods, of which the latter will be described in more detail in the remaining of
this chapter.
8.3.
FRAGMENT-BASED QUANTUM CHEMICAL APPROACHES
As just described, the long-range interactions have to be approximated to make
quantum chemical calculations for large molecules feasible. Here, we will concentrate on fragment-based approaches designed for macromolecular calculations. In
these methods, the large molecule is divided into small fragments and independent
quantum chemical calculations are performed on each fragment individually. If the
fragments are of similar size, and thus the computer time needed for each fragment is almost the same, a linear-scaling quantum chemical method can in principle
be obtained due to the fact that the number of fragments scales linearly with the
system size. In this way, Hartree-Fock as well as density functional theory (DFT)
calculations can be performed even for very large systems.
Probably the best-known fragment-based approach is the divide-and-conquer
ansatz by Yang, developed for semiempirical, HF, and density functional theory
methods [22–33]. Only the DFT version will be described here [23, 31], in which
the part of the electron density of a specific fragment, which contributes to the total
electron density of the large system, is calculated according to
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N
ρ k (r) = pk (r) ·
N
1
1 + e−β(eF −ei )
k
i
Cμi ϕμ (r) ,
μ
where the atomic orbital coefficients are taken from calculations of small molecules
composed of the fragment and additional surroundings in the same geometric
arrangement as in the macromolecule. pk (r) is a positive weighting function for
the subsystem k. This function has a large value near the fragment k and decreases
with increasing distance to the fragment. In this way, a specific fragment contributes
mainly to the region directly surrounding it exploiting the larger importance of shortrange interactions. The approximated total electron density is defined as the sum of
the electron density of the subsystems, from which the total energy of the system is
calculated according to standard density functional theory:
EKS (ρ) = TS (ρ) +
v(r)ρ(r)dr + EXP (ρ) +
1
2
ρ(r) · ρ(r )
drdr + VNN
r−r
The accuracy of the method compared to the direct calculation of the complete
molecule is determined by the amount of additional parts of the large molecule,
which are included in the fragment calculations, i.e. by the distance up to which
local interactions are included in the calculations of the subsystems.
The second approach, the adjustable density matrix assembler (ADMA)
[34–37] and its predecessor, the molecular electron density loge assemblers
(MEDLA) [38–40], were first introduced in the group of Mezey almost simultaneously to divide-and-conquer and then highly improved and extended in our group.
Since the discussion of long-range influences below will be based on this method,
it will be discussed in more detail using the Hartree-Fock level of theory here.
Analogously DFT can be applied [41]. ADMA combines fragment electron density matrices to approximate the electron density matrix of a large molecule (target
molecule) [34–37]. These fragment density matrices are taken from calculations of
smaller molecules (parent molecules) having a central region as one of the molecular fragments of the target macromolecule, surrounded by additional regions, called
surroundings in the following discussion, with the same local nuclear geometry as
in the macromolecule. These additional regions are taken within a selected distance
d of the central fragment (see Figure 8-1). Similar to divide-and-conquer, the accuracy of the ADMA method is solely controlled by the distance parameter d and,
by using a larger distance parameter, the results of direct quantum chemical calculations can be approximated with greater accuracy. In addition to the directly included
surroundings, the field-adapted ADMA (FA-ADMA) version [42, 43] approximates
the rest of the target macromolecule with partial charges in the parent molecule calculations. It was shown that the field-adapted approach leads to largely increased
accuracy with a fixed parent molecule size [42, 43].
An ADMA calculation is started with the subdivision of the target molecule into
a set of m mutually exclusive families of nuclei [34–37;42–45]. Standard quantum chemical calculations following the Hartree-Fock-Roothaan-Hall formalism are
Approximations of Long-Range Interactions
161
Figure 8-1. Visualization of the fragmentation process: the parent molecule for the red fragment is
defined as all atoms (in green) enclosed by a sphere of predefined radius. All other atoms are neglected
(ADMA) or included as partial charges (FA-ADMA, see below)
performed for the molecular fragments defined by one of these families, the corresponding quantum mechanical surroundings (defined by the distance parameter d),
and eventually also the partial charges at the positions of the rest of the nuclei. The
electron density ρ(r ) of a molecule can be expressed in terms of a basis set of n
atomic orbitals ϕi (r ) (i = 1,2,...,n) used for the expansion of the molecular wavefunction and the density matrix Pij determined for the given nuclear configuration
using this basis set:
n
n
Pij · ϕi (r ) · ϕj (r )
ρ(r ) =
i=1 j=1
Following the additive fuzzy density fragmentation (AFDF) scheme [35, 36], the
local fuzzy electron density fragments for one of the sets of mutually exclusive
families of nuclei denoted by f k , k = 1,...,m can be generated by defining a formal membership function mk (i) which indicates whether a given atomic orbital
(AO) basis function ϕi (r ) belongs to the set of AOs centered on a nucleus of the
family f k .
mk (i) =
1
0
if AO ϕi (r ) is centered on one nuclei of set f k
otherwise
Using this membership function, the elements Pkij of the density matrix of the kth
fragment are calculated according to the Mulliken-Mezey scheme [35, 36]:
Pkij = 0.5 · mk (i) + mk (j) · Pij
The membership function is designed so that the full and half matrix element value
is used, when both nuclei or only one nucleus belongs to the family f k , respectively.
If both nuclei belong to other families, a value of 0 is taken. The resulting fragment
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electron density matrices can then be summed up to get the total density matrix and
with that the total electron density of the large “target” macromolecule:
m
n
n
m
Pkij and ρ(r ) =
Pij =
k=1
Pkij · ϕi (r ) · ϕj (r )
i=1 j=1 k=1
From this, molecular properties, e.g. the total energy of the target molecule, can be
calculated following the standard Hartree-Fock formalism using the renormalized
ADMA approximation of the density matrix [42, 43, 45] instead of the ideal, directly
calculated exact macromolecular density matrix:
EHF =
1
2
n
n
m
Pkij + VNN
Fij + Hijcore ·
i=1 j=1
k=1
with:
b
b
(ij |rs ) −
Fij = Hijcore +
r=1 s=1
1
(is |rj ) ·
2
m
Pkij
k=1
Some other methods are based on the same idea as divide-and-conquer and ADMA
and will only be listed here: molecular tailoring approach of Gadre and coworkers
[46, 47], the central insertion scheme [48], and the fragment energy and localized molecular orbital assembler approach [49, 50]. A different combination of the
fragment calculations is applied in the fragment molecular orbital (FMO) method
[51–57]. Here, the electronic energy of the fragments and all pairs of fragments are
combined using the following equation:
E=
EIJ − EI − Ej +
I
J
EI
I
EIJ is the energy of the pair of fragments I und J and EI and EJ the energies obtained
from the single fragment calculations on fragment I and J, respectively. In newer
versions, energy gradients [54], molecule orbitals of the complete molecule [56],
and solvation effects [58] can also be calculated.
Last but not least, we would like to mention that also correlation methods can
profit from a fragment-based description. One such approach is the method of increments [59–62]. Like the other local correlation methods, the method of increments
starts with localized orbitals obtained from the Hartree-Fock wavefunction. It then
reduces the original correlation problem for the total number of electrons to a sum of
correlation contributions for smaller numbers of electrons, i.e. the correlation energy
is obtained by adding up correlation energy increments in terms of the localized
orbitals and in pairs, triples, . . . of the localized orbitals:
Approximations of Long-Range Interactions
Ecorr =
εI +
I
1
2
εIJ +
I
J=I
163
1
6
εIJK + · · ·
I
J=I K=I∨K=J
All size-extensive correlation methods can be used in this scheme to evaluate the
individual increments. Due to the local character of the orbitals it is also possible to
calculate the increments in finite embedded clusters, which correspond to the fragments and surroundings in the ADMA jargon. FMO calculations taking correlation
effects into account are also possible [63, 64].
8.4.
EMBEDDING SCHEMES
If the system is divided into two or more parts and, thus, the short-range and longrange interactions are described by two different theories as done in QM/MM as
well as fragment-based quantum chemical approaches, the interactions between the
subsystems have to be dealt with in a special way. In the QM/MM literature, these
special treatments are called embedding schemes and we will follow this nomenclature here due to the similarities between QM/MM and fragment-based methods. The
easiest possibility to combine the two parts (QM and MM region in QM/MM and
e.g. fragment plus surroundings and remaining part of the macromolecule in FAADMA) is the mechanical embedding scheme. Here, the QM calculations can be
performed totally independent on one of the parts without taking the other ones into
account, i.e. the calculation is performed on the QM part in vacuo without polarization due to the MM part (no point charges of the MM part are included as for the later
embeddings). In QM/MM, the QM and MM part interact by a Coulomb potential
between fixed point charges (for both the QM and MM part) and steric interactions
modeled by a van-der-Waals potential according to the force-field philosophy. The
Hamiltonian of the complete system can then be formulated as follows, employing
e.g. a Lennard-Jones potential for the van-der-Waals interactions and partial charges
taken from empirical force-fields:
Heff = HQM + HMM + HQM/MM
with
QM MM
HQM/MM =
i=1 j=1
qi · qj
+ 4εij
rij
σij12
rij12
−
σij6
rij6
In fragment-based methods, all interactions (steric as well as electrostatic as there
is no such strict separation between these two types) are calculated for the parent
molecules used in the individual QM calculations. Interactions between different
fragments are included since the atom sets defining the parent molecules overlap
but only as long as the atoms of one fragment are included in the surroundings of
the other fragment. In this way, short-range interactions are fully described by the
high level of theory but the long-range interactions are totally ignored.
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In the electric embedding scheme, molecular parts further away are included
as static point charges in the quantum mechanical calculations. Thus, the electron
density is polarized by these additional parts. This can be described by an effective
QM/MM Hamiltonian:
electrons MM
HQM/MM =
i=1
m=1
qm
+
rim
nuclei MM
k=1 m=1
Zk · qm
+ 4εkm
rkm
12
σkm
12
rkm
−
6
σkm
6
rkm
The interaction between the partial charges and the electron density as well as
between the nuclei is normally calculated in the QM and the van-der-Waals interaction in the MM program. For fragment-based calculations, the application of the
electric embedding is not that straightforward. The question here is how to define
the partial charges. One useful approximation is to also rely here on empirical
force fields and the partial charges defined in these, in which case the long-range
interactions are approximated with a lower level of theory.
In the last and most sophisticated approach, the polarized embedding scheme,
polarization effects for the complete molecule and all interactions are taken into
account. For doing so, polarizable force-fields, which are still subject of ongoing developments, have to be used in QM/MM calculations. Because these force
fields have a much higher computational demand, only a limited amount of studies have applied polarized embedding up to now. Full polarizability is much more
natural in fragment-based calculations and can be achieved with an iterative, selfconsistent cycle. First, a calculation with the mechanical embedding scheme is
performed. From this, partial charges are computed for the complete molecule
using e.g. Mulliken’s definition of partial charges. Then, each fragment calculation is repeated including the charges of the additional regions resulting in new
charges. This cycle is terminated if the charges do not change from one iteration to
the next.
8.5.
BORDER REGION
But even if one has decided on a specific embedding scheme, the combination of two
regions treated with different theories, where one is a quantum mechanical method,
causes an additional problem for (biological) macromolecules: If the system is
divided into subsystems, the border between these subsystems, almost unavoidably
in biomolecules, cuts through covalent bonds. For the QM calculation, it is not possible to simply truncate these bonds because this would lead to single-occupied
orbitals and would give strong perturbation of the electronic state. To circumvent
this problem different approaches were designed ranging from capping hydrogen
atoms over specially parameterized atoms or groups of atoms to localized hybrid
orbitals and generalized hybrid orbitals.
In most QM/MM and in almost all fragment-based studies, hydrogen atoms
are added to saturate dangling bonds. One and probably the most important reason for the widespread use of this approach is its striking simplicity. Since the