# Natural population analysis

15 Jul 1985-Journal of Chemical Physics (American Institute of PhysicsAIP)-Vol. 83, Iss: 2, pp 735-746

TL;DR: In this paper, a method of "natural population analysis" was developed to calculate atomic charges and orbital populations of molecular wave functions in general atomic orbital basis sets, which seems to exhibit improved numerical stability and to better describe the electron distribution in compounds of high ionic character.

Abstract: A method of ‘‘natural population analysis’’ has been developed to calculate atomic charges and orbital populations of molecular wave functions in general atomic orbital basis sets. The natural analysis is an alternative to conventional Mulliken population analysis, and seems to exhibit improved numerical stability and to better describe the electron distribution in compounds of high ionic character, such as those containing metal atoms. We calculated ab initio SCF‐MO wave functions for compounds of type CH3X and LiX (X=F, OH, NH2, CH3, BH2, BeH, Li, H) in a variety of basis sets to illustrate the generality of the method, and to compare the natural populations with results of Mulliken analysis, density integration, and empirical measures of ionic character. Natural populations are found to give a satisfactory description of these molecules, providing a unified treatment of covalent and extreme ionic limits at modest computational cost.

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TL;DR: The “Activation‐strain TS interaction” (ATS) model of chemical reactivity is reviewed as a conceptual framework for understanding how activation barriers of various types of reaction mechanisms arise and how they may be controlled, for example, in organic chemistry or homogeneous catalysis.

Abstract: We present the theoretical and technical foundations of the Amsterdam Density Functional (ADF) program with a survey of the characteristics of the code (numerical integration, density fitting for the Coulomb potential, and STO basis functions). Recent developments enhance the efficiency of ADF (e.g., parallelization, near order-N scaling, QM/MM) and its functionality (e.g., NMR chemical shifts, COSMO solvent effects, ZORA relativistic method, excitation energies, frequency-dependent (hyper)polarizabilities, atomic VDD charges). In the Applications section we discuss the physical model of the electronic structure and the chemical bond, i.e., the Kohn–Sham molecular orbital (MO) theory, and illustrate the power of the Kohn–Sham MO model in conjunction with the ADF-typical fragment approach to quantitatively understand and predict chemical phenomena. We review the “Activation-strain TS interaction” (ATS) model of chemical reactivity as a conceptual framework for understanding how activation barriers of various types of (competing) reaction mechanisms arise and how they may be controlled, for example, in organic chemistry or homogeneous catalysis. Finally, we include a brief discussion of exemplary applications in the field of biochemistry (structure and bonding of DNA) and of time-dependent density functional theory (TDDFT) to indicate how this development further reinforces the ADF tools for the analysis of chemical phenomena. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 931–967, 2001

8,490 citations

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TL;DR: Specific developments discussed include fast methods for density functional theory calculations, linear scaling evaluation of energies, NMR chemical shifts and electric properties, fast auxiliary basis function methods for correlated energies and gradients, equation-of-motion coupled cluster methods for ground and excited states, geminal wavefunctions, embedding methods and techniques for exploring potential energy surfaces.

Abstract: Advances in theory and algorithms for electronic structure calculations must be incorporated into program packages to enable them to become routinely used by the broader chemical community. This work reviews advances made over the past five years or so that constitute the major improvements contained in a new release of the Q-Chem quantum chemistry package, together with illustrative timings and applications. Specific developments discussed include fast methods for density functional theory calculations, linear scaling evaluation of energies, NMR chemical shifts and electric properties, fast auxiliary basis function methods for correlated energies and gradients, equation-of-motion coupled cluster methods for ground and excited states, geminal wavefunctions, embedding methods and techniques for exploring potential energy surfaces.

2,527 citations

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Yihan Shao, Zhengting Gan, Evgeny Epifanovsky

^{1}, Andrew T. B. Gilbert^{2}+153 more•Institutions (37)TL;DR: A summary of the technical advances that are incorporated in the fourth major release of the Q-Chem quantum chemistry program is provided in this paper, covering approximately the last seven years, including developments in density functional theory and algorithms, nuclear magnetic resonance (NMR) property evaluation, coupled cluster and perturbation theories, methods for electronically excited and open-shell species, tools for treating extended environments, algorithms for walking on potential surfaces, analysis tools, energy and electron transfer modelling, parallel computing capabilities, and graphical user interfaces.

Abstract: A summary of the technical advances that are incorporated in the fourth major release of the Q-Chem quantum chemistry program is provided, covering approximately the last seven years. These include developments in density functional theory methods and algorithms, nuclear magnetic resonance (NMR) property evaluation, coupled cluster and perturbation theories, methods for electronically excited and open-shell species, tools for treating extended environments, algorithms for walking on potential surfaces, analysis tools, energy and electron transfer modelling, parallel computing capabilities, and graphical user interfaces. In addition, a selection of example case studies that illustrate these capabilities is given. These include extensive benchmarks of the comparative accuracy of modern density functionals for bonded and non-bonded interactions, tests of attenuated second order Moller–Plesset (MP2) methods for intermolecular interactions, a variety of parallel performance benchmarks, and tests of the accuracy of implicit solvation models. Some specific chemical examples include calculations on the strongly correlated Cr_2 dimer, exploring zeolite-catalysed ethane dehydrogenation, energy decomposition analysis of a charged ter-molecular complex arising from glycerol photoionisation, and natural transition orbitals for a Frenkel exciton state in a nine-unit model of a self-assembling nanotube.

2,396 citations

01 Jan 2015

TL;DR: Detailed benchmarks of the comparative accuracy of modern density functionals for bonded and non-bonded interactions, tests of attenuated second order Møller–Plesset methods for intermolecular interactions, and tests of the accuracy of implicit solvation models are provided.

Abstract: A summary of the technical advances that are incorporated in the fourth major release of the Q-Chem quantum chemistry program is provided, covering approximately the last seven years. These include developments in density functional theory methods and algorithms, nuclear magnetic resonance (NMR) property evaluation, coupled cluster and perturbation theories, methods for electronically excited and open-shell species, tools for treating extended environments, algorithms for walking on potential surfaces, analysis tools, energy and electron transfer modelling, parallel computing capabilities, and graphical user interfaces. In addition, a selection of example case studies that illustrate these capabilities is given. These include extensive benchmarks of the comparative accuracy of modern density functionals for bonded and non-bonded interactions, tests of attenuated second order Møller–Plesset (MP2) methods for intermolecular interactions, a variety of parallel performance benchmarks, and tests of the accuracy of implicit solvation models. Some specific chemical examples include calculations on the strongly correlated Cr2 dimer, exploring zeolite-catalysed ethane dehydrogenation, energy decomposition analysis of a charged ter-molecular complex arising from glycerol photoionisation, and natural transition orbitals for a Frenkel exciton state in a nine-unit model of a self-assembling nanotube.

1,919 citations

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TL;DR: In this paper, the electronic structure of the radical CH 2 OH was analyzed via the "different hybrids for different spins" natural bond orbital (DHDS NBO) procedure, which finds separate Lewis structures for each of the spin systems.

Abstract: We have carried out ab initio UHF/6-31G* calculations on the hydroxymethyl radical, CH 2 OH, and have found the equilibrium structure to be nearly planar with barriers to internal rotation occurring at staggered and eclipsed geometries, in good agreement with experiment. The electronic structure of the radical was analyzed via the “different hybrids for different spins” natural bond orbital (DHDS NBO) procedure, which finds separate Lewis structures for each of the spin systems. The α spin Lewis structure resembles that of the anion; the β spin Lewis structure resembles the corresponding cation. This simple picture, in conjunction with Bent's rule, allows one to understand the principal electronic factors which dictate the structure of the radical CH 2 group and its torsional and inversion potentials. Charge transfer between oxygen non-bonding orbitals and the empty radical orbital in the β spin system is the dominant interaction determining the torsional potential. Smaller hyperconjugative interactions in the α spin system resemble interactions in closed-shell molecules and directly oppose the effect of radical hyperconjugation, thus illustrating the central idea that open-shell potential energy features result from competition between the two different spin systems.

1,866 citations

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TL;DR: In this paper, an analysis in quantitative form is given in terms of breakdowns of the electronic population into partial and total ''gross atomic populations'' and ''overlap populations'' for molecules.

Abstract: With increasing availability of good all‐electron LCAO MO (LCAO molecular orbital) wave functions for molecules, a systematic procedure for obtaining maximum insight from such data has become desirable. An analysis in quantitative form is given here in terms of breakdowns of the electronic population into partial and total ``gross atomic populations,'' or into partial and total ``net atomic populations'' together with ``overlap populations.'' ``Gross atomic populations'' distribute the electrons almost perfectly among the various AOs (atomic orbitals) of the various atoms in the molecule. From these numbers, a definite figure is obtained for the amount of promotion (e.g., from 2s to 2p) in each atom; and also for the gross charge Q on each atom if the bonds are polar. The total overlap population for any pair of atoms in a molecule is in general made up of positive and negative contributions. If the total overlap population between two atoms is positive, they are bonded; if negative, they are antibonded. Tables of gross atomic populations and overlap populations, also gross atomic charges Q, computed from SCF (self‐consistent field) LCAO‐MO data on CO and H2O, are given. The amount of s‐p promotion is found to be nearly the same for the O atom in CO and in H2O (0.14 electron in CO and 0.15e in H2O). For the C atom in CO it is 0.50e. For the N atom in N2 it is 0.26e according to calculations by Scherr. In spite of very strong polarity in the π bonds in CO, the σ and π overlap populations are very similar to those in N2. In CO the total overlap population for the π electrons is about twice that for the σ electrons. The most easily ionized electrons of CO are in an MO such that its gross atomic population is 94% localized on the carbon atom; these electrons account for the (weak) electron donor properties of CO. A comparison between changes of bond lengths observed on removal of an electron from one or another MO of CO and H2, and corresponding changes in computed overlap populations, shows good correlation. Several other points of interest are discussed.

9,238 citations

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TL;DR: In this paper, the effects of contraction on the energies and one-electron properties of the water and nitrogen molecules were investigated, and the authors obtained principles which can be used to predict optimal contraction schemes for other systems without the necessity of such exhaustive calculations.

Abstract: The contraction of Gaussian basis functions for use in molecular calculations is investigated by considering the effects of contraction on the energies and one‐electron properties of the water and nitrogen molecules. The emphasis is on obtaining principles which can be used to predict optimal contraction schemes for other systems without the necessity of such exhaustive calculations. Using these principles, contractions are predicted for the first‐row atoms.

4,595 citations

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TL;DR: In this article, a least square representation of Slater-type atomic orbitals as a sum of Gaussian-type orbitals is presented, where common Gaussian exponents are shared between Slater−type 2s and 2p functions.

Abstract: Least‐squares representations of Slater‐type atomic orbitals as a sum of Gaussian‐type orbitals are presented. These have the special feature that common Gaussian exponents are shared between Slater‐type 2s and 2p functions. Use of these atomic orbitals in self‐consistent molecular‐orbital calculations is shown to lead to values of atomization energies, atomic populations, and electric dipole moments which converge rapidly (with increasing size of Gaussian expansion) to the values appropriate for pure Slater‐type orbitals. The ζ exponents (or scale factors) for the atomic orbitals which are optimized for a number of molecules are also shown to be nearly independent of the number of Gaussian functions. A standard set of ζ values for use in molecular calculations is suggested on the basis of this study and is shown to be adequate for the calculation of total and atomization energies, but less appropriate for studies of charge distribution.

3,723 citations

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TL;DR: In this article, the authors define a set of generalized density matrices for the Hermitean density matrix of order $k, which is further antisymmetric in each set of these indices.

Abstract: In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of $N$ antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order $k$ has two sets of indices of each $k$ variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order $N$ over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete $k$-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly $N$ spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order $M$, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.

2,724 citations

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TL;DR: In this article, the authors show that the overlap integrals are of essential importance in molecules and in crystals, instead of being negligible, and the problem is simply solved by considering the orthonormalized functions [open phi]μ, given by (21), as the real atomic orbitals.

Abstract: The treatment of molecules and crystals by the Heitler‐London method or by the collective electron model can be based on the atomic orbitals φμ of the system. These orbitals are in general overlapping, and the corresponding overlap integrals Sμν, given by (1), have almost universally been neglected in the literature as causing undesirable complications. Here we will take these overlap integrals into consideration and show that they, instead of being negligible, are of essential importance in molecules and in crystals. The problem is simply solved by considering the orthonormalized functions [open phi]μ, given by (21), as the real atomic orbitals. The solution is worked out in detail for (I) the molecular orbital method of treating molecules, (II) the Bloch orbital method of treating crystals, and (III) the Heitler‐London method of treating both these systems in some simple spin cases. Some numerical applications are given for ionic crystals, showing that the overlap effects are responsible for all the rep...

2,702 citations