Chemical bonding in solids from ab-initio Calculations 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India & Center for Materials Science and Nanotechnology, University of Oslo, Norway http://folk.uio.no/ravi/cmt2015
Molecular Bonds Introduction 2 The bonding mechanisms in a molecule are fundamentally due to electric forces The forces are related to a potential energy function A stable molecule would be expected at a configuration for which the potential energy function has its minimum value
Features of Molecular Bonds 3 The force between atoms is repulsive at very small separation distances This repulsion is partially electrostatic and partially due to the exclusion principle Due to the exclusion principle, some electrons in overlapping shells are forced into higher energy states The energy of the system increases as if a repulsive force existed between the atoms The force between the atoms is attractive at larger distances
Potential Energy Function 4 The potential energy for a system of two atoms can be expressed in the form Ur A r B r () n m r is the internuclear separation distance m and n are small integers A is associated with the attractive force B is associated with the repulsive force
Potential Energy Function, Graph 5 At large separations, the slope of the curve is positive Corresponds to a net attractive force At the equilibrium separation distance, the attractive and repulsive forces just balance At this point the potential energy is a minimum The slope is zero
Bonds in Solids Types 6 Simplified models of bonding in solids include Ionic Covalent Metallic van der Waals Hydrogen
Ionic Bonding 7 Ionic bonding occurs when two atoms combine in such a way that one or more outer electrons are transferred from one atom to the other Ionic bonds are fundamentally caused by the Coulomb attraction between oppositely charged ions When an electron makes a transition from the E = 0 to a negative energy state, energy is released The amount of this energy is called the electron affinity of the atom The dissociation energy is the amount of energy needed to break the molecular bonds and produce neutral atoms
Ionic Bonding, NaCl Example 8 The graph shows the total energy of the molecule vs the internuclear distance The minimum energy is at the equilibrium separation distance
Ionic Bonding (final) 9 The energy of the molecule is lower than the energy of the system of two neutral atoms It is said that it is energetically favorable for the molecule to form The system of two atoms can reduce its energy by transferring electron out of the system and forming a molecule
Ionic Solids 10
Ionic Bonding 11 Transfer of electron from electropositive ion to the electronegative ion Coulomb interactions plays an important role Size of the cation reduces and size of anion increases Electro-negativity difference between ions decide the strength of ionic bonding.
Halite (NaCl) An example of Ionic Bonding 12
Electron Localization Function (ELF) for NaCl 13
Properties of Ionic Crystals 14 They form relatively stable, hard crystals They are poor electrical conductors They contain no free electrons Each electron is bound tightly to one of the ions They have high melting points They are transparent to visible radiation, but absorb strongly in the infrared region The shells formed by the electrons are so tightly bound that visible light does not possess sufficient energy to promote electrons to the next allowed shell Infrared is absorbed strongly because the vibrations of the ions have natural resonant frequencies in the low-energy infrared region
Covalent Bonding 15 A covalent bond between two atoms is one in which electrons supplied by either one or both atoms are shared by the two atoms Covalent bonds can be described in terms of atomic wave functions The example will be two hydrogen atoms forming H 2
Wave Function Two Atoms Far Apart 16 Each atom has a wave function 1 rao ψ1 s() r e 3 πa There is little overlap between the wave functions of the two atoms when they are far away from each other o
Combined Wave Functions 17 The wave functions can combine in the various ways shown Ψ s+ + ψ s+ is equivalent to ψ s+ + ψ s - These two possible combinations of wave functions represent two possible states of the two-atom system
Splitting of Energy Levels 18 The states are split into two energy levels due to the two ways of combining the wave functions The energy difference is relatively small, so the two states are close together on an energy scale For large values of r, the electron clouds do not overlap and there is no splitting of the energy level
Wave Function Covalent Molecule 19 The two atoms are brought close together The wave functions overlap and form the compound wave shown The probability amplitude is larger between the atoms than on either side
Sigma Bond Formation by Orbital Overlap 20 Two s orbitals overlap
Bonding in a Tetrahedron Formation of Hybrid Atomic Orbitals 21 4 C atom orbitals hybridize to form four equivalent sp 3 hybrid atomic orbitals.
Bonding in a Tetrahedron Formation of Hybrid Atomic Orbitals 22 4 orbitals in Si hybridize to form four equivalent sp 3 hybrid atomic orbitals.
Covalent Bonding (Final) 23 The probability is higher that the electrons associated with the atoms will be located between them This can be modeled as if there were a fixed negative charge between the atoms, exerting attractive Coulomb forces on both nuclei The result is an overall attractive force between the atoms, resulting in the covalent bond
Covalent Bonding 24
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ELF for Diamond 26
Electron Density of Si 27
Properties of Solids with Covalent Bonds 28 Properties include Usually very hard Due to the large atomic cohesive energies High bond energies High melting points Good thermal conductors
Cohesive Energies for Some Covalent Solids 29
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Metallic Solids 31 Metallic bonds are generally weaker than ionic or covalent bonds The outer electrons in the atoms of a metal are relatively free to move through the material The number of such mobile electrons in a metal is large
Metallic Solids, cont. 32 The metallic structure can be viewed as a sea or gas of nearly free electrons surrounding a lattice of positive ions The bonding mechanism is the attractive force between the entire collection of positive ions and the electron gas
Properties of Metallic Solids 33 Light interacts strongly with the free electrons in metals Visible light is absorbed and re-emitted quite close to the surface This accounts for the shiny nature of metal surfaces High electrical conductivity The metallic bond is nondirectional This allows many different types of metal atoms to be dissolved in a host metal in varying amounts The resulting solid solutions, or alloys, may be designed to have particular properties Metals tend to bend when stressed Due to the bonding being between all of the electrons and all of the positive ions
ELF for Na Metal 34 Electrons moved from ions and distributed uniformly in the interstitial regions. The electrons in the interstitial regions move freely around the crystal. Good conductors and ductile
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Another Carbon Example -- Buckyballs 36 Carbon can form many different structures The large hollow structure is called buckminsterfullerene Also known as a buckyball
Charge Density Distribution in TiC Mixed iono-covalent bonding 37 Depleted Charge at Ti site indicate the electron transfer from Ti to C ionic bonding Finite charge in between atoms and their nonspherical distribiution show the covalent bonding.
Diamond (covalent) Charge transfer plot 38 NaCl (ionic) Nb ( metallic)
Van der Waals Bonding 39 Two neutral molecules are attracted to each other by weak electrostatic forces called van der Waals forces Atoms that do not form ionic or covalent bonds are also attracted to each other by van der Waals forces The van der Waals force is due to the fact that the molecule has a charge distribution with positive and negative centers at different positions in the molecule As a result of this charge distribution, the molecule may act as an electric dipole Because of the dipole electric fields, two molecules can interact such that there is an attractive force between them Remember, this occurs even though the molecules are electrically neutral
Types of Van der Waals Forces 40 Dipole-dipole force An interaction between two molecules each having a permanent electric dipole moment Dipole-induced dipole force A polar molecule having a permanent dipole moment induces a dipole moment in a nonpolar molecule
Types of Van der Waals Forces, cont. 41 Dispersion force An attractive force occurs between two nonpolar molecules The interaction results from the fact that, although the average dipole moment of a nonpolar molecule is zero, the average of the square of the dipole moment is nonzero because of charge fluctuations The two nonpolar molecules tend to have dipole moments that are correlated in time so as to produce van der Waals forces
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Vander Waals Bonds Even in Solids 43
Hydrogen Bonding 44 In addition to covalent bonds, a hydrogen atom in a molecule can also form a hydrogen bond Using water (H 2 O) as an example There are two covalent bonds in the molecule The electrons from the hydrogen atoms are more likely to be found near the oxygen atom than the hydrogen atoms This leaves essentially bare protons at the positions of the hydrogen atoms The negative end of another molecule can come very close to the proton This bond is strong enough to form a solid crystalline structure
Hydrogen Bonding (Final) 45 The hydrogen bond is relatively weak compared with other electrical bonds Hydrogen bonding is a critical mechanism for the linking of biological molecules and polymers DNA is an example
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Polar and Nonpolar Bonds 47 AlP (polar) NaCl (ionic) Al (metallic) elf=0.5
Degree of Covalency From ELF 48
Compound with Semiconductivit y from metallic constituents (Cs 2 Pt) 49
Unusual Bonding (Charge density) LaNiInH 1.33 Formation of molecular like H-Ni-H (NiH 2 ) sub units 50
Unusual Bonding (Charge Transfer) LaNiInH 1.33 51 (r)= (r) comp. - (r) atomic Ionic bonding between H and the host lattice
The polarization of negative charges at H towards the electropositive La and In Unusual Bonding (ELF) LaNiInH 1.33 52
53 Charge density Charge transfer ELF Ravindran et al. PRL (2002)
Chemical bonding 54 KMgH 3 example for dominant ionic bonding Charge density Charge transfer ELF
La 3+ Sc 3+ O 3 6-55 Charge density Charge transfer ELF It is not a perfect ionic material and has finite covalency
One-dimensional Oxides 56 Crystal structure of Ca 3 Co 2 O 6 Hexagonal with 66 atoms /unit cell Co spins are arranged along c-axis For every two-up spin chain, there is one down-spin chain resulting in Ferrimagnetic structure Rhombohedral (R-3c space group) with 22 atoms/unit cell Alternating face-shared octahedra and trigonal one-dimensional chains along c-axis prisms form
Charge density and related features in Ca 3 Co 2 O 6 57 Charge density Charge transfer ELF Very weak interaction between Co1 and Co2 Magnitude of covalent interactions between Co1-O and Co2- O are different. There is strong covalent interaction within the chain and ionic interaction between Ca and the chains.
Bonding study from density of states analysis 58 Ca is in completely ionized state, donating almost all its valence electrons to O. Co is in two different oxidation and/or spin states, seen from difference in the topology of DOS curves. Oxygen is energetically degenerate with Co1 and Co2 forming strong covalent bond with them.
59 Partial Density Of States in KGaH 4 - Well separated Ga-s and -p state - Energetically degenerated H-s and Ga-p states - Strong covalent bonding Ga-H - Ionic bonding between K and H
60 Charge Density Distribution in KGaH 4 0.2 0.15 0.1 0.05 0.001
Visualization of lone pair From ELF 61 Lone pair electrons in BiMnO 3 Seshadri and Hill, Chemistry of Materials, 13, 2892 (2001)
62 ELF isosurface (0.7) show that Bi lone-pair electrons are responsible for ferroelectricity. Ravindran et al. PRB (2006)
63 Chemical bonding in BiFeO 3 from DOS analysis Isolated Bi-s is the lone pair electrons. Negligible Bi electrons in the valence band indicate ionic bonding Degenerate Fe-d and O-p states indicate the covalent bonding.
64 Charge density Charge Transfer ELF