Quantum crystallography: A perspective.

Extraction of the complete quantum mechanics from X-ray scattering data is the ultimate goal of quantum crystallography. This article delivers a perspective for that possibility. It is desirable to have a method for the conversion of X-ray diffraction data into an electron density that reflects the antisymmetry of an N-electron wave function. A formalism for this was developed early on for the determination of a constrained idempotent one-body density matrix. The formalism ensures pure-state N-representability in the single determinant sense. Applications to crystals show that quantum mechanical density matrices of large molecules can be extracted from X-ray scattering data by implementing a fragmentation method termed the kernel energy method (KEM). It is shown how KEM can be used within the context of quantum crystallography to derive quantum mechanical properties of biological molecules (with low data-to-parameters ratio). © 2017 Wiley Periodicals, Inc.

Single determinant N-representability and the kernel energy method applied to water clusters.

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single-determinant N-representable one-body density matrix. A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single-determinant N-representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N-representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.

Quantum Crystallography: Current Developments and Future Perspectives.

Crystallography and quantum mechanics have always been tightly connected because reliable quantum mechanical models are needed to determine crystal structures. Due to this natural synergy, nowadays accurate distributions of electrons in space can be obtained from diffraction and scattering experiments. In the original definition of quantum crystallography (QCr) given by Massa, Karle and Huang, direct extraction of wavefunctions or density matrices from measured intensities of reflections or, conversely, ad hoc quantum mechanical calculations to enhance the accuracy of the crystallographic refinement are implicated. Nevertheless, many other active and emerging research areas involving quantum mechanics and scattering experiments are not covered by the original definition although they enable to observe and explain quantum phenomena as accurately and successfully as the original strategies. Therefore, we give an overview over current research that is related to a broader notion of QCr, and discuss options how QCr can evolve to become a complete and independent domain of natural sciences. The goal of this paper is to initiate discussions around QCr, but not to find a final definition of the field.

Notes on The Energy Equivalence of Information.

Maxwell's demon makes observations and thereby collects information. As Brillouin points out such information is the negative of entropy (negentropy) and is the equivalent of a cost in energy. The energy cost of information can be quantified in the relationship E = kT ln 2, where k is the Boltzmann constant, T is the absolute temperature, and the factor ln 2 arises from the existence of two possibilities for a "yes/no" circumstance, as, for example, in the passage of a proton through a barrier controlled by a Maxwell's demon. This paper considers further conclusions that follow from the quantification of the energy cost of information. First, consideration of the minimum uncertainty in the measurement of energy cost of information leads to an expression for the uncertainty in the corresponding time of the measurement, which depends inversely upon temperature at which the measurements occur. Second, because of the universal connection between energy and mass, an almost imperceptible mass accompanies the accumulation of information. And third, to account for the total free energy change that describes the action of adenosine triphosphate (ATP) synthase, an additional term is suggested to be appended to the Mitchell chemiosmotic equation, which describes this process. The additional term accounts for the energy cost of sorting away from background ions those protons allowed to enter the ATP synthase.

Protoribosome by quantum kernel energy method.

Experimental evidence suggests the existence of an RNA molecular prebiotic entity, called by us the "protoribosome," which may have evolved in the RNA world before evolution of the genetic code and proteins. This vestige of the RNA world, which possesses all of the capabilities required for peptide bond formation, seems to be still functioning in the heart of all of the contemporary ribosome. Within the modern ribosome this remnant includes the peptidyl transferase center. Its highly conserved nucleotide sequence is suggestive of its robustness under diverse environmental conditions, and hence on its prebiotic origin. Its twofold pseudosymmetry suggests that this entity could have been a dimer of self-folding RNA units that formed a pocket within which two activated amino acids might be accommodated, similar to the binding mode of modern tRNA molecules that carry amino acids or peptidyl moieties. Using quantum mechanics and crystal coordinates, this work studies the question of whether the putative protoribosome has properties necessary to function as an evolutionary precursor to the modern ribosome. The quantum model used in the calculations is density functional theory--B3LYP/3-21G*, implemented using the kernel energy method to make the computations practical and efficient. It occurs that the necessary conditions that would characterize a practicable protoribosome--namely (i) energetic structural stability and (ii) energetically stable attachment to substrates--are both well satisfied.

Energy Equivalence of Information in the Mitochondrion and the Thermodynamic Efficiency of ATP Synthase.

Half a century ago, Johnson and Knudsen resolved the puzzle of the apparent low efficiency of the kidney (∼ 0.5%) compared to most other bodily organs (∼ 40%) by taking into account the entropic cost of ion sorting, the principal function of this organ. Similarly, it is shown that the efficiency of energy transduction of the chemiosmotic proton-motive force by ATP synthase is closer to 90% instead of the oft-quoted textbook value of only 60% when information theoretic considerations are applied to the mitochondrion. This high efficiency is consistent with the mechanical energy transduction of ATP synthase known to be close to the 100% thermodynamic limit. It would have been wasteful for evolution to maximize the mechanical energy transduction to 100% while wasting 40% of the chemiosmotic free energy in the conversion of the proton-motive force into mechanical work before being captured as chemical energy in adenosine 5'-triphosphate.

The localization-delocalization matrix and the electron-density-weighted connectivity matrix of a finite graphene nanoribbon reconstructed from kernel fragments.

Bader's quantum theory of atoms in molecules (QTAIM) and chemical graph theory, merged in the localization-delocalization matrices (LDMs) and the electron-density-weighted connectivity matrices (EDWCM), are shown to benefit in computational speed from the kernel energy method (KEM). The LDM and EDWCM quantum chemical graph matrices of a 66-atom C46H20 hydrogen-terminated armchair graphene nanoribbon, in 14 (2×7) rings of C2v symmetry, are accurately reconstructed from kernel fragments. (This includes the full sets of electron densities at 84 bond critical points and 19 ring critical points, and the full sets of 66 localization and 4290 delocalization indices (LIs and DIs).) The average absolute deviations between KEM and directly calculated atomic electron populations, obtained from the sum of the LIs and half of the DIs of an atom, are 0.0012 ± 0.0018 e(-) (∼0.02 ± 0.03%) for carbon atoms and 0.0007 ± 0.0003 e(-) (∼0.01 ± 0.01%) for hydrogen atoms. The integration errors in the total electron population (296 electrons) are +0.0003 e(-) for the direct calculation (+0.0001%) and +0.0022 e(-) for KEM (+0.0007%). The accuracy of the KEM matrix elements is, thus, probably of the order of magnitude of the combined precision of the electronic structure calculation and the atomic integrations. KEM appears capable of delivering not only the total energies with chemical accuracy (which is well documented) but also local and nonlocal properties accurately, including the DIs between the fragments (crossing fragmentation lines). Matrices of the intact ribbon, the kernels, the KEM-reconstructed ribbon, and errors are available as Supporting Information .

Calculation of strong and weak interactions in TDA1 and RangDP52 by the kernel energy method.

Using the Kernel Energy Method we apply ab initio quantum mechanics to study the relative importance of weak and strong interactions (including hydrogen bonds) in the crystal structures of the title compounds TDA1 and RangDP52. Perhaps contrary to widespread belief, in these compounds the weak interaction energies, because of their large number and cooperativity, can be significant to the binding energetics of the crystal, and thus also to its other properties.

Kernel energy method applied to vesicular stomatitis virus nucleoprotein.

The kernel energy method (KEM) is applied to the vesicular stomatitis virus (VSV) nucleoprotein (PDB ID code 2QVJ). The calculations employ atomic coordinates from the crystal structure at 2.8-A resolution, except for the hydrogen atoms, whose positions were modeled by using the computer program HYPERCHEM. The calculated KEM ab initio limited basis Hartree-Fock energy for the full 33,175 atom molecule (including hydrogen atoms) is obtained. In the KEM, a full biological molecule is represented by smaller "kernels" of atoms, greatly simplifying the calculations. Collections of kernels are well suited for parallel computation. VSV consists of five similar chains, and we obtain the energy of each chain. Interchain hydrogen bonds contribute to the interaction energy between the chains. These hydrogen bond energies are calculated in Hartree-Fock (HF) and Møller-Plesset perturbation theory to second order (MP2) approximations by using 6-31G** basis orbitals. The correlation energy, included in MP2, is a significant factor in the interchain hydrogen bond energies.

New structures of hydronium cation clusters.

Four new hydronium ion structures are investigated by means of quantum mechanical calculations at the DFT/B3LYP6-311+G(2d,2p) level of theory. There exist experimental crystallographic hydronium cations (H11O5 +) of two different geometrical structures, one BEXFEQ (acyclic) and one IYEPEH (cyclic). Molecular calculations reveal their relative stability. Another hydronium cation NEBDII (H15O7 +) when optimized reveals a totally new and unexpected structure. All three optimized structures are shown to be quite stable as judged by their binding energies, and therefore may possibly be found in solution. A main result of this article is the discovery of three new optimized structures of hydronium ions, all of which are preferentially ring structures. The optimized structure of H15O7 + is a cube lacking a vertex. Putting a water molecule at the "empty" vertex leads by energy optimization to a structure of H17O8 + which has the approximate symmetry of a cube. This cubic structure, as judged by its fragments, is one of the most interesting of the hydronium ions studied in this paper. The addition of H3O+ to a group of seven neutral molecules in the hypothetical reaction H3O+ + 7 H2O → H17O8 + induces two water molecules to each capture a proton at the expense of two other water molecules (converting them into hydroxyl anions) leading to a cluster with the formula [ H 3 O + 0.7 ] 3 [ H 2 O ] 3 + 0.1 [ OH - 0.6 ] 2 , where the superscripts are the integrated QTAIM atomic charges (in atomic units) on the respective species (inside the bracket) or on groups of a given species (outside the bracket). The cubic arrangement of 3H3O+.3H2O.2OH- is accompanied with a significant redistribution of charge: Each hydronium cation carries ca. +0.7 au, the hydroxyl anions only around -0.6 au each, while the water molecules remain quasi-neutral with a slight positive charge.

6-bromoindigo dye.

6-Bromoindigo (MBI) [systematic name: 6-bromo-2-(3-oxo-2,3-dihydro-1H-indol-2-ylidene)-2,3-dihydro-1H-indol-3-one], C(16)H(9)BrN(2)O(2), crystallizes with one disordered molecule in the asymmetric unit about a pseudo-inversion center, as shown by the Br-atom disorder of 0.682 (3):0.318 (3). The 18 indigo ring atoms occupy two sites which are displaced by 0.34 Šfrom each other as a result of this packing disorder. This difference in occupancy factors results in each atom in the reported model used to represent the two disordered sites being 0.08 Šfrom the higher-occupancy site and 0.26 Šfrom the lower-occupancy site. Thus, as a result of the disorder, the C-Br bond lengths in the disordered components are 0.08 and 0.26 Šshorter than those found in 6,6'-dibromoindigo (DBI) [Süsse & Krampe (1979). Naturwissenschaften, 66, 110], although the distances within the indigo ring are similar to those found in DBI. The crystals are also twinned by merohedry. Stacking interactions and hydrogen bonds are similar to those found in the structures of indigo and DBI. In MBI, an interaction of the type C-Br...C replaces the C-Br...Br interactions found in DBI. The interactions in MBI were calculated quantum mechanically using density functional theory and the quantum theory of atoms in molecules.

Characterization of a trihydrogen bond on the basis of the topology of the electron density.

Recent DFT calculations have predicted unexpected molecular structures for the ion induced dipole clusters H(n)(-) (3 ≤ n-odd ≤ 13). Analysis of these calculations suggests the definition of a new bond, called the trihydogen bond (THB). This is placed in context by a review and classification of multihydrogen interactions as usually discussed in the literature. The results of analysis related to the trihydrogen bond are presented. These include a series of linear relations exhibited by the H(n)(-) clusters involving the charge carried by the central H(-) ion, the binding energy of the clusters, and the relative stabilization of the central anion H(-) with respect to the energy of a free H(-) atom.

Ion induced dipole clusters H(n)- (3 ≤ n-odd ≤ 13): density functional theory calculations of structure and energy.

We investigate anew the possible equilibrium geometries of ion induced dipole clusters of hydrogen molecular ions, of molecular formula H(n)(-) (3 ≤ n-odd ≤ 13). Our previous publications [Sapse, A. M.; et al. Nature 1979, 278, 332; Rayez, J. C.; et al., J. Chem. Phys. 1981, 75, 5393] indicated these molecules would have a shallow minimum and adopt symmetrical geometries that accord with the valence shell electron pair repulsion (VSEPR) rules for geometries defined by electron pairs surrounding a central point of attraction. These earlier calculations were all based upon Hartree-Fock (HF) calculations with a fairly small basis of atomic functions, except for the H3(-) ion for which configuration interaction (CI) calculations were carried out. A related paper [Hirao, K.; et al., Chem. Phys. 1983, 80, 237] carried out similar calculations on the same clusters, finding geometries similar to our earlier calculations. However, although that paper argued that the stabilization energy of negative ion clusters H(n)(-) is small, vibration frequencies for the whole set of clusters was not reported, and so a definitive assertion of a true equilibrium was not present. In this paper we recalculate the energetics of the ion induced dipole clusters using density function theory (DFT) B3LYP method calculations in a basis of functions (6-311++G(d,p)). By calculating the vibration frequencies of the VSEPR geometries, we prove that in general they are not true minima because not all the resulting frequencies correspond to real values. By searching the energy surface of the B3LYP calculations, we find the true minimum geometries, which are surprising configurations and are perhaps counterintuitive. We calculate the total energy and binding energy of the new geometries. We also calculate the bond paths associated with the quantum theory of atoms in molecules (QTAIM). The B3LYP/6-311++G(d,p) results, for each molecule, deliver bond paths that radiate between each polarized H2 molecule and the polarizing H(-) ion.

The kernel energy method of quantum mechanical approximation carried to fourth-order terms.

It is now possible to calculate the ab initio quantum mechanics of very large biological molecules. Two things lead to this perspective, namely, (i) the advances of parallel supercomputers, and (ii) the discovery of a quantum formalism called quantum crystallography and the use of quantum kernels, a method that is well suited for parallel computation. The kernel energy method (KEM) carried to second order has been used to calculate the quantum mechanical ab initio molecular energy of peptides, protein (insulin and collagen), DNA, and RNA and the interaction of drugs with their biochemical molecular targets. The results were found to have good accuracy. In this article, the accuracy of the KEM is investigated up to an approximation including fourth-order interactions among kernels. Remarkable accuracy is achieved in the calculation of the energy of the ground state of the important biological molecule Leu1-zervamicin, whose crystal structure is known and used in the calculations.

Calculated interactions of a nitro group with aromatic rings of crystalline picryl bromide.

Several crystalline polymorphs have been discovered for picryl bromide. Among the several forces that control the formation of such polymorphs are the interactions among the nitro groups and phenyl rings of those crystals. Although there are >300 structures to be found in the Cambridge Structural Database displaying the nitro-phenyl interaction, nonetheless this interesting, and apparently important, interaction, seems not to have been discussed within any of the papers reporting the structures. In this article, quantum calculations are reported that assess the strength of these nitro-phenyl interactions within a crystal of picryl bromide. The rather flat molecules of picryl bromide are arranged in layered planes within the crystal, and we examine the attractive interactions that occur within a given plane, and between adjacent planes. Calculations of Hartree Fock and Møller Plesset perturbation theory carried to a second-order expansion are used. Both quantum mechanical approximations are implemented with 6-31G* basis functions.

A generalized higher order kernel energy approximation method.

We present a general mathematical model that can be used to improve almost all fragment-based methods for ab initio calculation of total molecular energy. Fragment-based methods of computing total molecular energy mathematically decompose a molecule into smaller fragments, quantum-mechanically compute the energies of single and multiple fragments, and then combine the computed fragment energies in some particular way to compute the total molecular energy. Because the kernel energy method (KEM) is a fragment-based method that has been used with much success on many biological molecules, our model is presented in the context of the KEM in particular. In this generalized model, the total energy is not based on sums of all possible double-, triple-, and quadruple-kernel interactions, but on the interactions of precisely those combinations of kernels that are connected in the mathematical graph that represents the fragmented molecule. This makes it possible to estimate total molecular energy with high accuracy and no superfluous computation and greatly extends the utility of the KEM and other fragment-based methods. We demonstrate the practicality and effectiveness of our model by presenting how it has been used on the yeast initiator tRNA molecule, ytRN(i)(Met) (1YFG in the Protein Data Bank), with kernel computations using the Hartree-Fock equations with a limited basis of Gaussian STO-3G type.

The Kernel Energy Method: Construction of 3 & 4 tuple Kernels from a List of Double Kernel Interactions.

The Kernel Energy Method (KEM) provides a way to calculate the ab-initio energy of very large biological molecules. The results are accurate, and the computational time reduced. However, by use of a list of double kernel interactions a significant additional reduction of computational effort may be achieved, still retaining ab-initio accuracy. A numerical comparison of the indices that name the known double interactions in question, allow one to list higher order interactions having the property of topological continuity within the full molecule of interest. When, that list of interactions is unpacked, as a kernel expansion, which weights the relative importance of each kernel in an expression for the total molecular energy, high accuracy, and a further significant reduction in computational effort results. A KEM molecular energy calculation based upon the HF/STO3G chemical model, is applied to the protein insulin, as an illustration.

Form factors for core electrons useful for the application of quantum crystallography (QCr) to organic molecules.

Form factors are calculated for the core electrons of the first-row atoms B, C, N, O and F. The form factors are presented in an analytical form, as appears in International Tables for X-ray Crystallography [Ibers & Hamilton (1974), Vol. IV, pp. 103-145. Birmingham: Kynoch Press; present distributor Kluwer Academic Publishers, Dordrecht]. Having such form factors for the core electrons reduces the number of parameters to be determined by the methods of quantum crystallography (QCr). It is shown that the form factors obtained are quite accurate. Thus, when they are used in QCr, they are expected to be consistent with accurate charge densities.

Wigner High Electron Correlation Regime in Nonuniform Electron Density Systems: Kinetic and Correlation-Kinetic Aspects.

The Wigner high electron correlation regime is characterized in the literature by an electron-interaction energy much greater than the kinetic energy. Via the 'quantal Newtonian' first law, we discover that for a nonuniform electron density system in this regime, there is a 'quantal compression' of the kinetic energy density. The explanation of this compression provides a fundamental understanding for why the kinetic energy is a smaller fraction of the total energy relative to the same ratio in the low correlation regime. We also discover by application of quantal density functional theory, that the contribution of electron correlations to the kinetic energy - the correlation-kinetic effects - and to the total energy is very significant. We propose that in addition to a high electron-interaction energy, the Wigner regime must thus also be characterized by a high correlation-kinetic energy.