Evolution of Quantum Chemistry

Notes

- Refer to Evolution of Quantum Theory

A second after Bing Bang [1], all particles scattered off each other at high rates and electrons would not bind with protons to form an atom. The photons being widely scattered off the electrically charged protons and electrons, could not travel far. Only when the universe expanded and temperature dropped to about 1 billion degree Kelvin, atomic nuclei existed, and at about 3000K, electrons and protons bounded to form a electrically neutral hydrogen atoms.

By 1000 BC, certain metals such as tin, lead, and copper were found to be used in making tools after recovering them from the ores simply by heating the rocks. The pre-Socratic philosopher, Empedociles around 420 BC stated that all matter is made up of four elements - earth, fire, air, and water, which was also supported by other philosophies including Vedic philosophy based on primary observable substances. Greek philosophers Leucippus and Democritus around 380 BC, proposed the atomic model based on indivisibility of atom ("not cut" in Greek) that reflects the properties of the matter as a whole. The idea was widely accepted before it was opposed by Greek Philosopher Aristotle in 330 BC who believed that all substances were made of small amounts of four elements of matter as showed in his scientific model (Fig. 1)

Fig. 1: Aristotle's Scientific Model consisting of 4 elements of matter

An early scientific method for chemistry emerged in 9th century initiated by chemists such as Jābir ibn Hayyān, who is known as the father of chemistry for his role in his systematic and experimental approach to scientific research based in the laboratory, and Rhazes, an early proponent of experimental medicine, who challenged Aristotle's theory of the four elements based on inexplicable properties of matter such as inflammability and salinity.

Robert Boyle, who is considered as the founder of modern chemistry presented Boyle's Laws in 1662 which describes the inverse proportionality between the absolute pressure and volume of a gas, given the temperature is kept constant within a closed system. He supported the views that the properties of matter are retained at the level of corpuscles (atoms). Hydrogen was isolated in 1776 by English chemist Henry Cavendish and oxygen was discovered in 1773 by Swedish chemist Carl Wilhelm Scheele which was isolated independently by English chemist Joseph Priestley in its gaseous state. The formation of dew (water) was first reported by Antoine-Laurent de Lavoisier in 1789 after combining hydrogen and oxygen. The first system of chemical nomenclature was devised by Lavoisier working with Claude Louis Berthollet and others.

Through Dalton's law that describes the relationship between the components in a mixture of gases and the relative pressure each contributes to that of the overall mixture, English meteorologist and chemist John Dalton in 1803 proposed a modern atomic theory in terms of law of multiple proportions - elements react in whole number multiples of discrete units of atoms and described three types of atoms - simple (elements), compound (simple molecules), and complex (complex molecules). The physicist Amedeo Avogadro in 1811 hypothesized that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. Later in 1865, Johann Josef Loschmidt determined the exact number of molecules in a mole, later named Avogadro's number.

In 1827, botanist Robert Brown used a microscope to look at dust grains floating in water and discovered that they moved about erratically, a phenomenon that became known as "Brownian motion". In 1905, Albert Einstein explained the motions by producing the first Statistical analysis of Brownian motion, and French physicist Jean Perrin used Einstein's explanation to experimentally confirm the atomic nature verifying Dalton's atomic theory.

The question of vitalism, the belief that living organisms are fundamentally different from non-living entity, was rejected after 1828 when Friedrich Wöhler synthesized urea, thereby establishing that organic compounds could be produced from inorganic starting materials.

Dmitri Mendeleev built up a systematic periodic table of all the 66 elements then known based on atomic mass, which he published in Principles of Chemistry in 1869. The table was compiled on the basis of arranging the elements in ascending order of atomic weight and grouping them by similarity of properties. With the discovery of the predicted elements, notably gallium in 1875, scandium in 1879, and germanium in 1886, it began to win wide acceptance. Later in 1913, Henry Moseley introduces the concept of atomic number to fix inadequacies of Mendeleev's periodic table.

The concept of free energy to explain the physical basis of chemical equilibria, the concept of free energy, now universally called Gibb's free energy, was contributed by Josiah Willard Gibbs in 1876.

Through the action of magnetic fields on the rays given out by the radium, that was isolated by the Curies couple in 1902, Pierre Curie proved the existence of particles electrically positive, negative, and neutral. These radioactive rays are later called alpha, beta, and gamma by Ernest Rutherford. Sir William Ramsay working with Frederick Soddy demonstrated the production of alpha particles (Helium nuclei) during the radioactive decay of radium.

With the discovery of electron in 1897 by J. J. Thompson when he was studying the properties of cathode rays setting up cathode ray tube as showed in Fig. 2, the atom no longer remained the smallest thing that existed. The deflected cathode rays or electrons towards positive electric plate were turned out to be 2000 times lighter than hydrogen.

Fig. 2: J.J. Thomson's experiment set-up

Thomson therefore proposed a plum pudding model of an atom in 1904 as showed in Fig. 3 where negatively charged electrons, are distributed about a positively charged medium like a plums in plum pudding that would balance out overall charges to form a neutral atom. In 1909, physicist R. Millikan measured the charge of an electron using Oil Drop Experiment.

Fig. 3: J.J. Thomson's Plum Pudding Model [2]

Thomson plum pudding model was disproved by Ernest Rutherford in 1911 after interpreting gold foil experiment (see Fig. 4) performed by his students in 1909 which is called Rutherford model.

Fig. 4: A beam of Alpha particles scattering off gold atoms

Most of the alpha particles did not deviate, some deflected slightly, and few deviated with higher angles of 90 -180 degrees as showed in Fig. 4 indicating that atoms are mostly empty space, the nucleus is positive, and contain these charged particles in small region that occupies most of the atomic mass. The electrons orbit the nucleus by their velocity like the planet orbiting the sun. The physicists realized that the atomic model offered by Rutherford was not complete as mass of nucleus was experimentally found to be approximately twice than the number of protons. Rutherford did postulate the existence of some neutral particle having mass similar to proton but there was no direct experimental evidence. It however led to the discovery of isotopes by the chemist Francis William Aston in 1921 and then the neutron by the physicist James Chadwick in 1932 working with Rutherford. The models for a nucleus composed of protons and neutrons were developed within months after the discovery of the neutron by Dmitri Ivanenko and Werner Heisenberg.

The centripetal acceleration experienced by the orbiting charged electrons would produce electromagnetic waves, lose energy, and spiral into nucleus. Hence, a Danish physicist Niels Bohr improved Rutherford's model by proposing that electrons traveled about the nucleus in orbits that had specific energy levels applying Planck's quantum idea to a problem in atomic physics in 1913. So, the angular momentum of electrons is restricted by:

L = mvr = nh/2π

where "h" is the Planck's constant and "n" is the integer that represents the principle quantum number which gives the radius "r" of the allowed energy states as showed in Fig. 5.

Fig. 5: Bohr's model showing different allowed energy levels for electrons. An electrons jumps to different orbit accompanied by an emitted or absorbed amount of electromagnetic energy "hf" called quanta [2].

Bohr model had circular orbits for electrons but Arnold Sommerfeld revised his model introducing another quantum number, azimuthal quantum number (l) in 1915, for electrons to take as many shapes corresponding to the shell number (principle quantum number n) with the maximum number of electrons in each shape to be calculated by:

2(2l + 1), where l = n-1

In the first shell (n = 1) the orbital shape is given by 0 ( 1 - 1) i.e l = 0 which is circular and only 2 electrons can occupy that orbit. The shapes become more complex with the higher shell number as showed in Fig. 6.

Diagram showing the shapes of electron orbitals.

Fig. 6: Orbital shapes in different shells [5]

Sommerfeld in 1920 realized that the orbital shapes can have different orientation in space and he introduced magnetic quantum number (ml) to explain the Zeeman effect (Fig. 7), the splitting of spectral lines in the presence of magnetic field.

Fig. 7: Zeeman Effect on Hydrogen Atom [6]

The maximum number of different orientation can be found using the formula:

2l + 1

For example, for l = 1, there are three possible orientations ml = -l to l i.e. -1, 0, and 1 as showed in Fig. 8.

Diagram showing different orientations for each possible shape.

Fig. 8: Different orientations of Orbital shape [5]

The 1924 paper from Edmund C. Stoner, helped Wolfang Pauli realized that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state, if the electron states are defined using four quantum numbers to describe various atomic spectra. He devised the quantum mechanical principle, Pauli Exclusion Principle for electrons in 1925 that states that no two electrons can share the same quantum state at the same time i.e. i.e. no two electrons in a single atom can have the same n, l, ml, and ms numbers. Though he showed that electron states in an atom can be described by four quantum numbers (n, l, ml, ms) each quantum number corresponds to a degree of freedom of the electron, he could not explain physical significance for the fourth quantum number, which was needed empirically.

While studying certain details of spectral lines known as the anomalous Zeeman effect (zeeman effect with electron spin) as in Fig. 9 (a), George Uhlenbeck and Samuel Goudsmit in 1925 realized that Pauli’s fourth quantum number must relate to electron spin, an intrinsic angular momentum independent of its orbital angular momentum.

Fig. 9 (a): Greater Variety of splitting patterns on zinc triplet not observable in zinc singlet where net angular momentum is zero, that is, overall spin quantum number ms=0 [7].

The magnetic field can separate the degenerated state - linearly independent Eigen state (energy levels) with the same Eigen value (energy), into separate energy levels that results in spectral lines of slightly different energies (anomalous Zeeman effect). Due to the spin pairing based on Pauli Exclusion Principle, most molecules do not exhibit magnetic field i.e. spin quantum number (S) is zero and are diamagnetic. When one electron is excited to a higher energy level as showed in Fig. 9b, it can be promoted in the same spin orientation as it was in the ground state (antiparallel) and is called the singlet excited state. In a triplet excited state, the promoted electron has the same spin orientation (parallel) as the other unpaired electron. The triplet state has total spin quantum number S = 1 and is paramagnetic. The spin multiplicity value and its corresponding spin state was first discovered by Friedrich Hund in 1925 and Hund's rule gives the maximum spin-multiplicity for triplet state as 2S + 1 = 2*1 + 1 = 3.

Fig. 9(b) Spin in the Ground and Excited State - Singlet and Triplet Excited States [16]

A molecule can have a singlet state or triplet state with different energy and both states can inter-convert by a process called inter-system crossing. Phosphorescence and fluorescence are based on this principle.

After the confirmation of wave-particle duality of electron, De Broglie reinterpreted Bohr's restriction of angular momentum through the concept of standing waves in 1924 where wavelength of electron must fit into the atomic space being bounded to an atom as showed in Fig. 10.

Image result for De Broglie's re-interpretation of Bohr's condition

Fig. 10: De Broglie's re-interpretation of Bohr's condition of allowed orbits; (a) standing wave (b) wavelength of electron as a wave fitting into the circumference (allowed orbit) (c) wavelength not fitting into the circumference (not allowed orbit) [4]

In 1926, Erwin Schrödinger used De Broglie's idea to develop a mathematical model of the atom that described the electrons as three-dimensional waveforms rather than point particles. The solutions of Schrödinger equation give rise to three quantum numbers associated with the energy levels of the atom that could predict many of the spectral phenomena that Bohr's model failed to explain. Max Born formulated the interpretation of the probability density function for ψ*ψ in the Schrödinger equation in 1926 that described probability density of electron waves, the idea that the electrons exist in a superpositional 'probability cloud', a fuzzy cloud spread over the entire atom that represents the probability of finding an electron at any particular region around the nucleus, and its pattern is referred to as atomic orbitals. Considering electron as a waveform, it is mathematically impossible to simultaneously derive the position and momentum of an electron according to Heisenberg uncertainty principle formulated in 1926 which invalidated the Bohr's model of well-defined orbitals for electrons.

Pauli found that the non-relativistic Schrödinger equation had incorrectly predicted the magnetic moment of hydrogen to be zero in its ground state and solved the nonrelativistic theory of spin in 1927. This became the basis for the derivation of relativistic quantum mechanics (RQM) by Paul Dirac in 1928. According to this theory, force carrier photon carries away the lost energy from electron jump to preserve the conservation of energy.

The electrons in an atom are attracted to the protons in the nucleus by the electromagnetic force which binds the electrons inside an electrostatic potential well surrounding the smaller nucleus as showed in Fig. 11 for hydrogen. The closer an electron is to the nucleus, the greater the attractive force. Hence electrons bound near the center of the potential well require more energy to escape than those at greater separations. A long-range attraction that tends to move the electron closer to the nucleus, and a short-range effect that tends to spread the electron out more around the nucleus balances at the electronic ground state (n = 1), the minimum energy state of electrons in an atom or zero point.

Fig. 11: Potential energy, V(x), in a hydrogen atom and first three probability densities with l = 0. The probability densities are shifted by the corresponding electron energy [8]

Edward Frankland, F.A. Kekulé, A.S. Couper, Alexander Butlerov, and Hermann Kolbe developed the theory of valency by the mid 19th century which was built on the theory of radicals. Valency was called "combining power", in which compounds were joined owing to an attraction of positive and negative poles. A bonding theory (chemical bonding) based on the number of electrons in the outermost "valence" shell of the atom was explained by Gilbert Newton Lewis with the concept of cubic atoms with their outer electrons in the corner following the cycle of 8 elements in the periodic table. His book "The Atom of the Molecule" published in 1916 suggested that a chemical bond is a pair of electrons shared by two atoms introducing the electron dot diagram which is known as Lewis structure. In the same year, Walther Kossel put forward a theory assuming complete transfer of electrons between atoms based on Abegg's rule formulated by Richard Abegg in 1904. Irving Langmuir popularized and elaborated Lewis's model introducing the term covalent bond between 1919 and 1921.

Walter Heitler working with his associate formulated the Heitler–London theory in 1927 to determine the covalent bond between hydrogen atoms joining their wavefunctions using quantum mechanics of Schrödinger's wave equation. It was extended by John C. Slater and Linus Pauling to Valence-Bond (VB) or Heitler–London–Slater–Pauling (HLSP) method. The theory explains the geometry of the compound by the overlap of half-filled valence atomic orbitals of each atom containing one unpaired electron.

Molecular Orbital Theory, less intuitive to chemist, that determines the molecular structure based on mathematical functions of moving electrons (delocalized) under the influence of the nuclei in the whole molecule rather than localized to individual bonds between atoms, was first introduced by Friedrich Hund and Robert S. Mulliken in 1927 and 1928. A molecular orbital theory describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. It results in a more complete picture of the structure of a chosen molecule.

An exact solution for the Schrödinger equation can only be obtained for the hydrogen atom. The repeated interactions between particles create quantum correlations, or entanglement, and their Schrödinger equations cannot be solved exactly. The wave function of the system becomes a complicated object holding a large amount of information, which usually makes exact or analytical calculations impractical or even impossible. So, approximate solutions were sought for these many body problems.

In 1927 D. R. Hartree introduced the self-consistent field (SCF) method, iterative method where an an initial set of orbitals is used to generate a new set of orbitals with convergence criteria (e.g. number of SCF cycles), to calculate approximate wave functions and energies for atoms and ions. It is the simplest approximation that follows that the electrons are independent, and interact only via the mean-field Coulomb potential, and hence yields one-electron Schrödinger equation. Table [1] illustrates how much the number of SCF cycles varies with the SCF convergence criterion for the HF/STO-3G calculation on formaldehyde. HF/STO-3G & SCF=(Conver=n) are defined in Gaussian input file. It can be clearly seen that the total energy of the system is converged to within 10-9 Hartree already with SCF=(Conver=5) in 7 SCF cycles. For details, visit Gaussian Tutorial.

Table 1: Variation of SCF Cycles with SCF Convergence Criteria

In 1930, Slater and V. A. Fock independently pointed out that the Hartree method did not respect the Pauli's principle of antisymmetry of the wave function. This exchange condition was satisfied by forming a Slater determinant of the individual orbits resulting in Hartree-Fock method that follows Born-OppenHeimer (BO) approximation. BO approximation was proposed in 1927 by Born and Oppenheimer that allows waveform of molecule to be broken into its electronic and nuclear (vibrational, rotational) components, clamping the nuclei at certain position in space taking into account their interactions with electrons. Such interactions are represented by single scalar potentials called Potential Energy Surface (PES) or adiabatic surface (Fig. 12).

Fig. 12: PES for water; minimum energy (minima) at O-H bond length of 0.0958nm and H-O-H bond angle of 104.5° [2].

Here, energy is the function of two co-ordinates - bond length and bond angle. If there is only one coordinate, the surface is called a potential energy curve or energy profile (or reaction coordinate diagram) such as Morse potential.

The saddle points in PES corresponds to transition states, the highest point on the reaction coordinates and is the lowest energy pathway connecting a chemical reactant to a chemical product. Such concept of PES surface for chemical reactions was first suggested by the French physicist René Marcelin in 1913. Fig. 13 shows how the enzyme is lowering the activation energy in a chemical reaction.

Fig. 13: Catalytic Reaction on the Reaction Coordinate [2]

The correlational energy in DFT that is treated as a functional of electronic density which is the function of the position, considerably reduces the computational complexity. To make up for the accuracy taking into account exchange and correlation effects, Kohn-Sham method is followed to map the problem of the system interacting electrons onto a fictitious system of non-interactive electrons. The local (spin-) density approximation (LDA) is simple resulting in a realistic description of the atomic structure, elastic, and vibrational properties for a wide range of system. But it overestimates the binding energies of molecules and solids. So, Generalized Graident Approximations (GGA) has been introuduced to overcome such deficiencies.

Linear combinations of atomic orbitals (LCAO), approximation for molecular orbitals introduced in 1929 by Sir John Lennard-Jones, can be used to estimate the molecular orbitals that are formed upon bonding between the molecule's constituent atoms. Molecular orbitals are obtained from the combination of atomic orbitals - sums and differences of atomic waveforms, which predict the location of an electron in an atom specifying the electron configuration of a molecule as showed in Fig. 14.

Fig. 14: Hydrogen Molecule created by computing mathematical functions for the two 1s orbitals i.e. σ symmetry. Electrons spend most of their time in region between two nuclei in bonding orbitals and away from that region in anti-bonding orbitals [9]

The bonding interactions are in-phase or constructive whereas antibonding interactions are out-of-phase or destructive, and nobonding with a nodal plane where the wavefunction of the antibonding orbital is zero between the two interacting atoms.

The molecule is more stable in the bonding region as showed in Fig. 15.

Fig. 15: Bonding and anti-bonding Energy [9]

The two electrons associated with a pair of hydrogen atoms are placed in the lowest energy, or bonding, molecular orbital, as shown in Fig 16.

Fig. 16: The energy of the hydrogen molecule is lower than that of pair of isolated atoms i.e. essential condition to form a molecule which explains why there is no He2 molecule i.e. He atoms have non-bonding interactions [9]

As showed in the Molecular diagram and orbital energy diagram of O2 in Fig. 17, the core orbitals on an atom make no contribution to the stability of the molecule that contain this atom.

Fig. 17: Molecular and orbital energy diagrams of O2 [9]

Fig 17 shows significant difference between the energies of the 2s and 2p orbitals; the head-on interaction between the 2pz orbitals is stronger than the edge-on interaction between the 2px or 2py orbitals. Hence, the 2p orbital lies at a lower energy than the πx and πy orbitals, and the 2p* orbital lies at higher energy than the πx* and πy* orbitals. There are also higher-order delta (δ) and phi (φ) symmetries.

In O2, there is no interaction between 2s and 2p orbitals which is possible for the molecules such as N2 which introduces an element of s-p mixing, or hybridization, into the molecular orbital theory. The result is a slight change in the relative energies of the molecular orbitals as showed in Fig 18.

Fig. 18: Hybridization in N2; 2s orbital on one atom to interact with the 2pz orbital on the other [9]

N2 having eight electrons in bonding orbitals and two electrons in antibonding orbitals, has a bond order of three, which constitutes a triple bond as two electrons in an antibonding molecular orbital cancel the effect of one bond.

The qualitative approach to molecular orbital theory through LCAO is part of the start of modern quantum chemistry where the orbitals are described as linear combination of basic functions which are typically one-electron functions (atomic orbitals) centered on the nuclei of the component atoms of the molecule. In mathematical sense, these wave functions describes electrons of a given atom using set of such basic functions called basis set. The atomic orbitals like hydrogen-like atoms known analytically Slater Type Orbitals (STOs) was first introduced by John C Slater in 1930 with e-r radial decay moving away from nucleus. The difficulty in analytical ingratiation i.e. computational difficulty of STO led to its approximation through Gaussian Type Orbitals (GTOs) by Frank Boys in 1950s as showed in Fig. 19.

There are hundred of basis sets composed of GTOs and GTO-3G is the simplest atomic orbital representation, and is called a minimal basis set. In the extended basis sets, there are more basic functions for each Atomic Orbital (AO). Basic sets are called double-zeta (DZ), triple-zeta (TZ), quadruple-zeta(QZ), 5Z, and so on depending on the number of basic functions for each AO. Split-valence basis set uses only one basis function for core and more basis functions for Valence AO. In Pople basis set, a contracted basis set developed by John Pople, 3-21G is a split valence basis set where core orbitals are a contraction on 3 GTO’s, the inner part of the valence AOs is a contraction of 2 GTO’s and the outer part is given by 1 GTO.

Fig. 19: STO (Solid line) and GTO-3G (STO approximated by 3 GTOs)

With the atoms interacting, atom's orbital may get shifted to one side or the other i.e. atoms get polarized. So, the most common addition to minimal basis sets is the polarization function. Pople basis set 6-31G* [or 6-31G(d)] is 6-31G with added "d" polarization functions on non-hydrogen atoms; 6-31G** [or 6-31G(d,p)] is 6-31G* plus "p" polarization functions for hydrogen. To accurately represent the "tail" portion of the atomic orbitals, which are distant from the atomic nuclei, diffuse functions are added. It is mandatory for computation of anions. Pople basis set 6-31+G is 6-31G plus defuse s and p functions for non-hydrogen atoms; 6-31++G also has diffuse functions for hydrogen.

The processes that involve electronic rearrangements such as chemical reactions cannot be described at the Molecular Mechanics (MM) level. At the same time, the computational demand for evaluating the electronic structure places severe constraints on the size of the system that can be studied. Ab Inito methods (e.g. Hartree-Fock and DFT (Density Functional Theory)) mostly based on Perturbation Theory and semi-empirical methods (e.g. CNDO/2, MINDO, AM1, PM3) based on Quantum Mechanics (QM) along with force fields (e.g. MM2,AMBER, CHARMM, GROMACS) in Molecular Mechanics (MM) or combination of both i.e. QM/MM are used in the computation model to provide an approximate potential energy in Potential Energy Surface (PES). Though both Ab Inito and semi-empirical methods make approximations, semi-empirical methods obtain some parameters from empirical data to improve the calculation time which makes them good fit for larger molecules (organic chemistry) where only a few elements are used extensively, and Ab Inito methods are too expensive. Semi-empirical methods are based on the Hückel method, the first simple LCAO-MO method called Hückel Molecular Orbital Theory (HMO) proposed by Erich Hückel in 1930s for pi-orbitals. It was extended to conjugated molecules considering all valence electrons by Ronald Hoffman in 1963.

Force Fields uses functional form of potential energy in molecular mechanics. It was first introduced into chemistry by Hill and by Westheimer in 1949 and includes both the bonded (e.g covalent bond ) and non-bonded (e.g. electrostatic (ionic bond), van der walls) terms of inter or intra molecular forces (molecular interactions). The covalent bond forms by the sharing of electrons and includes the terms bond-length (stretching), bond-angle, and dihedral (rotation along the bond) as showed in Fig. 20. Covalent bonds with enthalpies on the order of 100 kcal/mole [19] are stronger than molecular interactions with enthalpy 1-10 kcal/mol between a pair of non-bonded atoms. In a chemical reaction, the covalent bonds form or break but they remains intact during molecular break down or physical change such as protein folding/unfolding, separation of DNA strands, changing state of water (ice, vapor).

Fig. 20: Interactions between bonded atoms [10]

The electrostatic interaction between pair of charged atoms is represented by Coulomb potential. The van der walls interaction between two atoms arise from a balance (minimum V* at the optimal separation rm) between repulsive and attractive forces as represented by Lennard-Jones potential (Fig. 21 a) which was proposed in 1924 by John Lennard-Jones. When the atoms are closer i.e. r is small electron-electron interaction (repulsion) is strong but when the atoms are farther apart, the fluctuation in charge distribution induces dipoles in atoms or molecules resulting in attractive dispersion forces (e.g. London dispersion (Fig. 21 b), dipole-dipole (Fig. 22). At a considerably large atomic separation r, there will be no interaction i.e. V=0. Some force field also includes explicit terms of hydrogen bonds as showed in Fig. 23. The electronic interactions in NaCl due to considerable difference in electronegativity in Na and Cl form strong ionic bonds.

Fig. 21 (a): Van der walls interaction modeled using Lennard-Jones 6-12 potential [2]

Fig. 21: London Dispersion; unsymmetrical distribution of electrons inducing temporary dipole (with δ+ and δ- partial charges)

Fig. 22 (a): The positive-end (δ+) of Polar molecule Formaldehyde (permanent dipole) attracting to the negative-end (δ-)of another molecule.

Fig. 23: Hydrogen bond formation with highly electronegative atom Nitrogen.

Fig. 24: Electrostatic Interaction - ion pair within a folded protein; anionic aspartic acid engages with cationic arginine; the dashed lines showing hydrogen bonds [19]

The force field hence defines set of parameters for bonded and non-bonded terms as showed thorough the potential function for CHARMM in Fig. 25.

Fig. 25: CHARMM potential function [10].

The geometry of the molecule determines many of its physical and chemical properties. The objective of geometric optimization is to find the best atomic arrangement that makes the molecules most stable. So, it begins with the input structure that is believed to resemble the desired one and the computer algorithm like Gaussian, CHARMM systematically changes the geometry until the convergence criteria is met. Geometric optimization characterizes the stationary points (minimum, transition state or higher order saddle point) on a Potential Energy Surface (PES). The Intrinsic Reaction Coordinate (IRC) path in PES connects reactants to the products which are both energy minima, passing through a saddle point known as transition state. Once the stationary point has been found by geometric optimization, we need to determine whether it is a minima, transitional state, or higher order saddle point by calculating the vibrational frequencies. The structure is a minimum or the most stable if the imaginary frequency is zero. Multiple minimal points can correspond to different conformations or structural isomers. So, for finding the global minima, we need to compare the energy for different isomers as showed in Fig. 26.

Fig. 26: PES for Butane conformational isomers - Eclipsed Syn, Staggered Gauche, Eclipsed, and Staggered Anti from left to right obtained through Coordinate scan done at B3LYP/6-311+G(2d,p) using WebMO[11]

In molecular simulation, the most time consuming part of an energy or force computation is the evaluation of the LJ and electrostatic interactions among all pairs of particles. The cutoff function in CHARMM sets the cut-off for electrostatic and LJ interactions modulated by switching or shift functions which is included in CHARMM all-atom force field as [12] which is portrayed in Fig. 27.

ENERgy SHIFt VSWItch CTOFnb 12. CTONnb 10

Fig. 27. The electrostatic options are shifted to zero using a cut-off value of 12 angstroms (CTOFnb) and that LJ interactions are modulated by a switching function between 10 (CTONnb) and 12 angstroms (CTOFnb). Any interactions beyond 12 angstroms are discarded [12].

The cut-off alone is not sufficient for simulating the infinite systems of bulk gasses, liquids, crystals or mixtures. Periodic Boundary Condition (PBC) therefore approximates them using exact copies of the small part called unit cell (Fig. 28). In CHARMM's terminology these are the image boxes (cubic simulation box). Ewald summation/Particle-Mesh-Ewald (PME) is used to calculate electrostatic interaction under PBC.

Fig. 28: The square shape represents the unit cell for infinite system in 2D which can be repeated throughout the system.

To precisely emulate the biological systems through simulations that usually occur in solution, water molecules seen to be embedded in pockets during NMR and X-Ray crystallography, the proteins are solvated or hydrated by placing them either in a water box using PBC or in a water sphere in surrounding vacuum without PBC as showed in Fig. 29.

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Fig. 29: Hydrated Ubiquitin Protein (a) in a water box (b) in a water sphere [14]

Indeed, proteins lack biological activity in the absence of sufficient hydrating water. Protein folding, the coiling of proteins into a specific three-dimensional shape, is driven by its interaction with water as it emerges from ribosomal synthesis into the the bulk aqueous phase of the cytoplasm. The potential energy funnel for the folding of proteins without sufficient water present in Fig. 30 (a) [13] highlights barriers to the preferred minimum energy structure on the folding pathway which can trap the protein in an inactive three dimensional molecular conformations. When the protein is fully hydrated the potential energy is seen to be considerably smoothed as as showed in Fig 30 (b) which allows proteins to attain their active minimum-energy conformations in a straightforward and rapid manner.

Fig. 30: Potential Energy funnel for a protein (a) without sufficient water and (b) with sufficient water [13]

Water molecules also form an integral part of most protein-protein, protein-DNA, and protein-ligand interactions. In Figure 31, drug (ligand) is binding to specific protein (receptor) through docking in water medium.

Fig. 31. Protein-Ligand Interaction in the medium of water [13]

During the chemical reaction, the total energy of the system (Enthalpy (H)) can change, and depending on the nature of the change, it can be exothermic (ΔH < 0 i.e. gives off heat) or endothermic (ΔH > 0 i.e. absorb heat) reactions. Besides exothermic reactions, there are other reactions that increases the disorder (Entropy (S)) of the system and are spontaneous. The favorability of the reaction is determined by Gibbs free energy (G) of the system that has both favorable and unfavorable driving forces. For the constant temperatures, the change is Gibbs energy is given by:

G = H - TS

During folding or unfolding of proteins, the free energy change (ΔG) is due to the combined effects of both protein folding/unfolding and hydration changes. These compensate to such a large extent that the free energy of stability of a typical protein is only 40-90 kJ mol-1 (equivalent to very few hydrogen bonds), whereas the enthalpy change and temperature times the entropy change may be greater than ±500 kJ mol-1 different. There are both enthalpic and entropic contributions to this free energy that change with temperature and so give rise to heat denaturation and, in some cases, cold denaturation due to breakup of the water network around the protein [13].

The atoms in a molecule are constantly in motion and the entire molecule experiences constant transnational and rotational motion represented by 3N degree of freedom where N is the number of atoms. For non-linear water molecule, the motion of molecule in relation to the the cartesian co-ordinates has three transnational and three rotational motions totalling 6 motions, and remaining 3N - 6 i.e. 3*3 - 6 = 3 vibrational motions (complex vibrations of poly-atomic molecule resolved into Normal Modes - asymmetric, symmetric, and bending (wagging, twisting, scissoring, and rocking)). IR spectrum of H20 in Fig. 31 shows three bands - asymmetric stretch at 3756 cm-1, symmetric stretch at 3657 cm-1 and the bending motion (scissors band) at 1595 cm-1. Diatomic molecules having single vibration are only observed in Raman Spectra but not in IR Spectra.

Fig. 31: Three Vibrational motions (Normal Modes) of a water molecule picked up by Infra Red (IR) Spectroscopy i.e. IR active [15]

In solids, bands result from the overlap of atomic orbitals similar to molecular orbitals (MO) result from the overlap of atomic orbitals (AO) in small molecules. The orbitals spread over many atoms and blend into a band of molecular orbitals. The range of energies of these orbitals are so closely spaced that instead of having discrete energies as in the case of free atoms, the available energy states form bands. The occupied bands of bonding orbitals (by electrons) are called energy bands or allowed bands or Valence band (HOMO band) while the unallowed ones next to them are called forbidden or band gaps which is followed by the highest empty band of antibonding orbitals called conduction band (LUMO band). In a metal, the band of energy levels is only partly filled. The highest filled level right before going to the empty level (band gap) is called the Fermi level or electrochemical potential.

The probability of the energy state with certain energy being occupied by an electron at thermodynamic equilibrium is given by Fermi-Dirac Distribution (1926) which was used by Sommerfeld in metals in 1927. Fermi-Dirac distribution defined the Fermi Level where the state has 50% chances of being occupied. In 1928, Fowler and Nodheim suggested that electrons could come from a single band occupied in accordance with Fermi-Dirac distribution (statistics) or could be emitted in statistically different ways under different conditions of temperature and applied field.

Band Theory based on Quantum Mechanical (QM) and Density Functional Theory (DFT) derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. The complexity is built on free electron model using properties of potential that is periodic on a lattice defined by Block theorem developed by Felix Bloch in 1928. The wave momentum vector k (electron state) or orbital with energy E(k) appearing in Bloch's theorem can always be confined to the first Brillouin zone (primitive cell of a crystal) as showed in Fig. 32. A band structure can be computed by solving the Kohn-Sham equation for many different k points along different lines of the Brillouin zone.

Fig. 32: (a) Energy E of free electrons plotted vs K (b) Periodic zone space in K-space (c) Shape of the energy spectrum when confining the consideration to the first Brillouin zone [18]

Green's function of a system provides both ground (the total energy) and also excited state observables of the system. The poles of the Green's function are the quasiparticle energies, the bands of a solid, and it can be calculated by solving Dyson equations with GW self-energy approximations. Band Theory can explain many physical properties of solids including electrical resistivity and optical absorption forming the foundation of the understanding solid-state devices such as transistors and solar cells. In semi-conductor materials, the Fermi level lies inside the band gap but the bands are much closer to the Fermi level to be thermally populated with electrons or holes as showed in Fig. 33.

Fig. 33: Fermi level EF (dotted line) and filling of electronic states in various type of materials.The shade follows the Fermi–Dirac distribution (black = all states filled, white = no state filled) [2]

In Metals, the valence electrons from the s and p orbitals of the interacting metal atoms delocalize forming a "sea" of electrons that surrounds the positively charged atomic nuclei of the interacting metal ions. The free electrons among a lattice of positively charged metal ion results in a strong attractive force between them forming a metallic bonds (Fig. 32) which imparts metallic traits such as strength, malleability, ductility, luster, conduction of heat and electricity - the broken local bonds can get easily reformed (strength, malleability, ductility), energy can pass quickly through free electrons (good conductor), and light can't easily penetrate the densely packed atoms and get reflected (luster).