^{1,a)}, C. Di Paola

^{1}and L. Kantorovich

^{1,b)}

### Abstract

We show that, at least for the ground electronic state of systems treated using semilocal density functionals (like in local density or generalized gradient approximations), a calculation of the entire extended nonperiodic system consisting of several well distinguished parts (e.g., a collection of molecules) can be replaced with a finite set of calculations on specifically chosen smaller subsystems that overlap with each other. Every subsystem is terminated with link (or pseudo) atoms (or groups of atoms) to reduce the effect of the termination. However, because of the particular choice of the subsystems, the effect of the link atoms is largely compensated in the final total energy if the subsystems are chosen sufficiently large. In fact, we prove that the proposed method should result in nearly the same total energy, electronic density and atomic forces as a single (considered as a reference) density functional calculation on the entire system. Our method, however, should be much more efficient due to unfavorable scaling of the modern electronic structure methods with the system size. The method is illustrated on examples of serine water, lysine-water and lysine dimer systems. We also discuss possible approximate applications of our method for quantum-classical calculations of extended systems, when, as compared to widely used quantum-mechanical/molecular-mechanical methods, the problem of the quantum cluster boundary can be eliminated to a large degree.

J. He and C. Di Paola would like to acknowledge the financial support from the European Integrated project PicoInside (Contract No. FGP-015847). We would also like to thank G. A. DiLabio for providing us with detailed explanations on the usage of the H-like pseudopotential.

I. INTRODUCTION

II. THEORY

A. Additivity rule

B. Partitioning scheme

C. Implementation and efficiency

III. RESULTS

A. Serine-water system

B. Lysine-water system

C. Lysine dimer

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Density functional theory
- 25.0
- Total energy calculations
- 25.0
- Macromolecules
- 7.0
- Biomolecular interactions
- 6.0
- Ab initio calculations
- 5.0

## Figures

Schematics of two molecules and , and their partitioning into six regions: contact , buffer ( and ), and remote ( and ).

Schematics of two molecules and , and their partitioning into six regions: contact , buffer ( and ), and remote ( and ).

Auxiliary systems enabling calculation of the total energy of the entire system: (a) subsystems and ; (b) subsystem (the “contact region”), and (c) subsystems and . Small solid circles correspond to possible LAs.

Auxiliary systems enabling calculation of the total energy of the entire system: (a) subsystems and ; (b) subsystem (the “contact region”), and (c) subsystems and . Small solid circles correspond to possible LAs.

Serine-water system: (a) geometry and atomic Mulliken charges after the full system geometry relaxation; all atoms in the pair are marked, and the charges (in electrons) are given in parenthesis; (b) subsystem containing water and a part of the serine molecule terminated with the LA; (c) subsystem containing only the part of the serine molecule of subsystem , also terminated with the same LA. (d) Comparison of system geometries relaxed using our partitioning method (bright) and when considering the hole system as one (dim). Oxygen atoms are red, H atoms white, C atoms green, and N atoms blue, while the LA is yellow.

Serine-water system: (a) geometry and atomic Mulliken charges after the full system geometry relaxation; all atoms in the pair are marked, and the charges (in electrons) are given in parenthesis; (b) subsystem containing water and a part of the serine molecule terminated with the LA; (c) subsystem containing only the part of the serine molecule of subsystem , also terminated with the same LA. (d) Comparison of system geometries relaxed using our partitioning method (bright) and when considering the hole system as one (dim). Oxygen atoms are red, H atoms white, C atoms green, and N atoms blue, while the LA is yellow.

Lysine-water: (a) geometry and Mulliken charges (in brackets) on atoms obtained in the DFT calculation in which the whole system was considered as one; (b) the 13 atoms subsystem containing the LA at the position of carbon atom of the lysine molecule; (c) the corresponding subsystem; (d) a bigger 19 atoms subsystem , also terminated with the LA replacing atom ; and (e) the corresponding subsystem . (f) Two relaxed geometries of the lysine-water system, superimposed on each other: one obtained using our method with the smallest subsystem (bright) and the other when treating the pair as a single system (dim). The coloring system used is the same as in Fig. 3.

Lysine-water: (a) geometry and Mulliken charges (in brackets) on atoms obtained in the DFT calculation in which the whole system was considered as one; (b) the 13 atoms subsystem containing the LA at the position of carbon atom of the lysine molecule; (c) the corresponding subsystem; (d) a bigger 19 atoms subsystem , also terminated with the LA replacing atom ; and (e) the corresponding subsystem . (f) Two relaxed geometries of the lysine-water system, superimposed on each other: one obtained using our method with the smallest subsystem (bright) and the other when treating the pair as a single system (dim). The coloring system used is the same as in Fig. 3.

Relative total energies of the lysine-water system as functions of the distance (in Å) between the two molecules (see insets where only the relevant part of the lysine molecule is shown) for three (chosen arbitrarily, see text) orientations of the water molecule, calculated using the exact method (solid lines) and our partitioning scheme (dashed lines). Note that geometry relaxation was not performed in these calculations. The total energies include the Coulomb correction at each point. These were however found to be very small (less than 0.02 eV). The zero energy corresponds to the exact energy minimum obtained with the exact method.

Relative total energies of the lysine-water system as functions of the distance (in Å) between the two molecules (see insets where only the relevant part of the lysine molecule is shown) for three (chosen arbitrarily, see text) orientations of the water molecule, calculated using the exact method (solid lines) and our partitioning scheme (dashed lines). Note that geometry relaxation was not performed in these calculations. The total energies include the Coulomb correction at each point. These were however found to be very small (less than 0.02 eV). The zero energy corresponds to the exact energy minimum obtained with the exact method.

Lysine dimer: (a) geometry of the dimer and Mulliken charges (in brackets) on atoms obtained in the DFT calculation in which the whole system was considered as one; (b) the smallest 20 atoms subsystem containing LAs on both sides at the positions of carbon atom of both molecules; (c) the corresponding subsystem (the subsystem, not shown, is built similarly from the other molecule); (d) a bigger 32 atoms subsystem , also terminated with two LAs at the position of and (e) the corresponding subsystem . (f) Two relaxed geometries of the lysine dimer, one obtained using our method with the smallest subsystem (partition 1) and the other when treating the dimer as a single system, superimposed on each other. The coloring system used is the same as in Fig. 3. Because of the symmetry, numbering of atoms and atomic charges are mainly shown only for one molecule for simplicity.

Lysine dimer: (a) geometry of the dimer and Mulliken charges (in brackets) on atoms obtained in the DFT calculation in which the whole system was considered as one; (b) the smallest 20 atoms subsystem containing LAs on both sides at the positions of carbon atom of both molecules; (c) the corresponding subsystem (the subsystem, not shown, is built similarly from the other molecule); (d) a bigger 32 atoms subsystem , also terminated with two LAs at the position of and (e) the corresponding subsystem . (f) Two relaxed geometries of the lysine dimer, one obtained using our method with the smallest subsystem (partition 1) and the other when treating the dimer as a single system, superimposed on each other. The coloring system used is the same as in Fig. 3. Because of the symmetry, numbering of atoms and atomic charges are mainly shown only for one molecule for simplicity.

A schematic comparison of the MFCC partitioning scheme (a) with ours (b) for the case of a single molecule divided into two fragments and . In the MFCC method, three artificial systems are considered: , , and the conjugated caps system . In our method five smaller subsystems are employed: (which is the fragment with a simple termination), ( with a termination), and , , and , the latter being essentially a superposition of and . Different semiovals designate the corresponding simple terminations (e.g., hydrogens) in both methods.

A schematic comparison of the MFCC partitioning scheme (a) with ours (b) for the case of a single molecule divided into two fragments and . In the MFCC method, three artificial systems are considered: , , and the conjugated caps system . In our method five smaller subsystems are employed: (which is the fragment with a simple termination), ( with a termination), and , , and , the latter being essentially a superposition of and . Different semiovals designate the corresponding simple terminations (e.g., hydrogens) in both methods.

## Tables

Comparison of angles (in degrees) and interatomic distances (in Å) in the lysine-water system calculated by (i) applying the DFT method to the whole system (second column) and (ii) using our method with both partitioning schemes. The results of our method are shown as deviations from the reference values in the second column and are presented in the third and fourth columns.

Comparison of angles (in degrees) and interatomic distances (in Å) in the lysine-water system calculated by (i) applying the DFT method to the whole system (second column) and (ii) using our method with both partitioning schemes. The results of our method are shown as deviations from the reference values in the second column and are presented in the third and fourth columns.

Comparison of Mulliken charges on atoms of the lysine-water system obtained using our partition method with subsystem containing 13 atoms (partitioning method 1), shown for different subsystems in columns 3–5, and after the reference calculation (the last column) in which the whole system was considered as one. The first column shows atom number according to Fig. 4(a), regions within molecule are shown in the second column. The charges without brackets obtained for subsystems and are used in our method as final atomic charges; the charges in the brackets do not represent the charges on atoms; however, they are needed to calculate the Coulomb correction, see Eq. (25).

Comparison of Mulliken charges on atoms of the lysine-water system obtained using our partition method with subsystem containing 13 atoms (partitioning method 1), shown for different subsystems in columns 3–5, and after the reference calculation (the last column) in which the whole system was considered as one. The first column shows atom number according to Fig. 4(a), regions within molecule are shown in the second column. The charges without brackets obtained for subsystems and are used in our method as final atomic charges; the charges in the brackets do not represent the charges on atoms; however, they are needed to calculate the Coulomb correction, see Eq. (25).

Comparison of angles (in degrees) and interatomic distances (in Å) in the lysine dimer calculated by (i) applying the DFT method to the whole system (second column), (ii) using our partitioning method 1 (third column) and (iii) method 2 (fourth column). The results of our method are shown as deviations from the reference values in the second column and are presented in the third and fourth columns. Both the results calculated with the PBE and B3LYP (in brackets) functionals are shown.

Comparison of angles (in degrees) and interatomic distances (in Å) in the lysine dimer calculated by (i) applying the DFT method to the whole system (second column), (ii) using our partitioning method 1 (third column) and (iii) method 2 (fourth column). The results of our method are shown as deviations from the reference values in the second column and are presented in the third and fourth columns. Both the results calculated with the PBE and B3LYP (in brackets) functionals are shown.

Comparison of Mulliken charges on atoms of the lysine dimer obtained using our partition method with subsystem containing 20 atoms (partitioning method 1), shown for different subsystems in columns 3–5, and after the exact calculation (the last column) in which the dimer was considered as one system. The PBE density functional was used. The first column shows atom number according to Fig. 6(a), regions within molecule are shown in the second column. Charges on atoms of the second molecule, subsystem , are not shown since these are the same on corresponding atoms due to symmetry. The charges without brackets obtained for subsystems and are used in our method as final charges on the atoms of the dimer; the charges in the brackets do not represent the charges on the dimer atoms; however, they are needed to calculate the Coulomb correction, Eq. (17).

Comparison of Mulliken charges on atoms of the lysine dimer obtained using our partition method with subsystem containing 20 atoms (partitioning method 1), shown for different subsystems in columns 3–5, and after the exact calculation (the last column) in which the dimer was considered as one system. The PBE density functional was used. The first column shows atom number according to Fig. 6(a), regions within molecule are shown in the second column. Charges on atoms of the second molecule, subsystem , are not shown since these are the same on corresponding atoms due to symmetry. The charges without brackets obtained for subsystems and are used in our method as final charges on the atoms of the dimer; the charges in the brackets do not represent the charges on the dimer atoms; however, they are needed to calculate the Coulomb correction, Eq. (17).

Comparison of Mulliken charges on atoms of the lysine dimer obtained using our partition method with subsystem containing 32 atoms (partitioning method 2) and after the exact calculation. See the caption to Table IV for further details.

Comparison of Mulliken charges on atoms of the lysine dimer obtained using our partition method with subsystem containing 32 atoms (partitioning method 2) and after the exact calculation. See the caption to Table IV for further details.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content