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6 Multi -- Procrustes Fragmentation with Trigonometric Weighting

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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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large molecules. Chem Phys Lett 424:420–424

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

6 Multi -- Procrustes Fragmentation with Trigonometric Weighting