Chemistry
Molecular Orbital Theory
Molecular Orbital Theory is a model used to describe the behavior of electrons in molecules. It involves the combination of atomic orbitals to form molecular orbitals, which can be bonding, non-bonding, or anti-bonding. This theory provides a more accurate description of molecular structure and bonding compared to the simpler valence bond theory.
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12 Key excerpts on "Molecular Orbital Theory"
eBook - PDF- David R. Klein(Author)
- 2016(Publication Date)
- Wiley(Publisher)
1.8 Molecular Orbital Theory 17 1.8 Molecular Orbital Theory In most situations, valence bond theory will be sufficient for our purposes. However, there will be cases in the upcoming chapters where valence bond theory will be inadequate to describe the observa- tions. In such cases, we will utilize Molecular Orbital Theory, a more sophisticated approach to viewing the nature of bonds. Molecular orbital (MO) theory uses mathematics as a tool to explore the consequences of atomic orbital overlap. The mathematical method is called the linear combination of atomic orbitals (LCAO). According to this theory, atomic orbitals are mathematically combined to produce new orbitals, called molecular orbitals. It is important to understand the distinction between atomic orbitals and molecular orbitals. Both types of orbitals are used to accommodate electrons, but an atomic orbital is a region of space associated with an individual atom, while a molecular orbital is associated with an entire molecule. That is, the molecule is considered to be a single entity held together by many electron clouds, some of which can actually span the entire length of the molecule. These molecular orbitals are filled with electrons in a particular order in much the same way that atomic orbitals are filled. Specifically, electrons first occupy the lowest energy orbitals, with a maximum of two electrons per orbital. In order to visualize what it means for an orbital to be associated with an entire molecule, we will explore two molecules: molecular hydrogen (H 2 ) and bromomethane (CH 3 Br). Consider the bond formed between the two hydrogen atoms in molecular hydrogen. This bond is the result of the overlap of two atomic orbitals (s orbitals), each of which is occupied by one electron. According to MO theory, when two atomic orbitals overlap, they cease to exist. Instead, they are replaced by two molecular orbitals, each of which is associated with the entire molecule (Figure 1.15).
eBook - PDF- John Lowe(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
CHAPTER 14 QUALITATIVE Molecular Orbital Theory 14-1 The Need for a Qualitative Theory Ab initio and semiempirical computational methods have proved extremely useful. But also needed is a simple conceptual scheme that enables one to predict the broad outlines of a calculation in advance, or else to rationalize a computed result in a fairly simple way. Chemistry requires conceptual schemes, simple enough to carry around in one's head, with which new information can be evaluated and related to other information. Such a theory has developed along-side the mathematical methods described in earlier chapters. We shall refer to it as qualitative Molecular Orbital Theory (QMOT). In this chapter we describe selected aspects of this many-faceted subject and illustrate QMOT applications to questions of molecular shape and conformation, and reaction stereochemistry. 14-2 Hierarchy in Molecular Structure and in Molecular Orbitals We seek a simple qualitative approach to the question, How does the total energy of a system change as we move the nuclei with respect to each other? This question is very broad, encompassing the phenomena of molecular structure and chemical reactivities. It is useful to distinguish three kinds of process that can occur as nuclei are moved. One of these is the process in which two nuclei move closer together or farther apart, with their separation being somewhere around 1 or 2 A (i.e., about one bond length) either at the outset or the conclusion of the motion (or both). This process includes the breaking or forming of bonds and also the stretching or compressing of bonds. It also includes the forcing together of two species that will not bond (e.g., He with He). Let us refer to this as a nearest-neighbor interaction, even though the two nuclei need not be bonded in the usual chemical sense. The second process is the changing of the bond angle between two nuclei bonded to a third. The changing of the H-O-H angle in water is an example.
eBook - PDF- John Lowe(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
CHAPTER 14 QUALITATIVE Molecular Orbital Theory 14-1 The Need for a Qualitative Theory Ab initio and semiempirical computational methods have proved extremely useful. But also needed is a simple conceptual scheme that enables one to predict the broad outlines of a calculation in advance, or else to rationalize a computed result in a fairly simple way. Chemistry requires conceptual schemes, simple enough to carry around in one's head, with which new information can be evaluated and related to other information. Such a theory has developed along- side the mathematical methods described in earlier chapters. We shall refer to it as qualitative Molecular Orbital Theory (QMOT). In this chapter we describe selected aspects of this many-faceted subject and illustrate QMOT applications to questions of molecular shape and conformation, and reaction stereochemistry. 14-2 Hierarchy in Molecular Structure and in Molecular Orbitals We seek a simple qualitative approach to the question, "How does the total energy of a system change as we move the nuclei with respect to each other?" This question is very broad, encompassing the phenomena of molecular structure and chemical reactivities. It is useful to distinguish three kinds of process that can occur as nuclei are moved. One of these is the process in which two nuclei move closer together or farther apart, with their separation being somewhere around 1 or 2Ä (i.e., about one bond length) either at the outset or the conclusion of the motion (or both). This process includes the breaking or forming of bonds and also the stretching or compressing of bonds. It also includes the forcing together of two species that will not bond (e.g., He with He). Let us refer to this as a nearest- neighbor interaction, even though the two nuclei need not be bonded in the usual chemical sense. The second process is the changing of the bond angle between two nuclei bonded to a third. The changing of the H-O-H angle in water is an example.
eBook - PDFQuantum Nanochemistry, Volume Three
Quantum Molecules and Reactivity
- Mihai V. Putz(Author)
- 2016(Publication Date)
- Apple Academic Press(Publisher)
22 Quantum Nanochemistry—Volume III: Quantum Molecules and Reactivity • The number of involved functions ϕ i immediately becomes very large and therefore has, also without involving the excited orbital (states), about 4 millions for the simple ion as SO 4 − , for example; • Using truncating method (as in Heitler and London method – to be amended in the Section 1.4) and then applying the perturbation approx-imations (viz. adiabatic coupling), leads to uncertain procedures. For all these reasons, soon after the mesomerism, Hund and Mulliken had been developed a more consistent theory with the physical signifi -cance of molecular orbitals, which confers the atoms as the rooting role of molecular orbitals’ formation, from where the molecular orbitals (MO) method was as such nominated. 1.3 Molecular Orbital Theory OF BONDING The basic idea is simple: next to each nucleus, the most representative terms are those who correspond to the smallest nucleus-electron distances, so the total Hamiltonian is a little different respecting that of the isolated atom. Therefore the wave-functions which describe the electrons around each atom of bonding, called atomic orbitals , represent approximate local solutions of the Schrödinger’s equation in molecule; accordingly, one will search for the molecular function, called molecular orbital , as a linear combination of the atomic orbitals, or hybridized, of various atoms get-ting into the molecule (Hückel, 1934; Coulson, 1938, 1952; Hall, 1950; Griffith & Orgel, 1957; Jensen, 1999; Licker, 2004). Resuming, the chemical bond which may be formed between two atoms A and B is considered as the resulted MO/the wave-function by “composition” of the two atomic orbitals ϕ A and ϕ B which by overlap-ping/superposition constitute the bond Ψ AB A B c c = + 1 2 ϕ ϕ (1.62) without other approximations.
eBook - PDF- David V. George(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
10 Molecular-orbital Theory 10-1 INTRODUCTION So far in this book we have considered only very simple, idealized systems like the rigid rotator and the harmonic oscillator, and the very simplest of systems of real chemical interest, the hydrogen and helium atoms. We had to start at this level of simplicity in order to present the fundamental ideas of quantum mechanics, and also to indicate how to set about getting exact solutions to quantum mechanical problems. But the real use of quantum mechanics to chemists is in the help it can give us when we try to understand molecular properties. We are especially interested in the nature of chemical bonding; we are also interested in explaining theoretically experimental results such as spectroscopic measurements, dipole moments and ionization potentials. In addition, we would like to be able to make calculations of molecular properties that are experimentally inaccessible —the molecules might have too transient an existence, or the conditions of temperature and pressure make it impossible to get reliable experimental results. In a book of this type we cannot cover all the applications of quantum mechanics in chemistry. We shall, however, treat some of the most important; in parti-cular we shall examine the nature of molecular bonding, and most of the rest of the book is devoted to this question. Unfortunately, molecular systems are generally far too complex for us to attempt to obtain exact solutions to their quantum mechanical equations. Though computers now allow us to do molecular calculations that used to be impossibly lengthy, we still have to resort to many approximations. We expect, therefore, that the results may not always 169 170 Molecular-orbital Theory be in very good agreement with experiment. Nevertheless, they cer-tainly encourage us to believe that quantum mechanics gives a true description of molecular systems.
eBook - PDF- Harry G. Brittain(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
PART III. VALENCE ELECTRON SPECTROSCOPY 4 Molecular Orbital Theory and the Electronic Structure of Molecules Harry G. Brittain Center for Pharmaceutical Physics, Milford, New Jersey, U.S.A. MOLECULAR ORBITAL (MO) THEORY OF ORGANIC MOLECULES In the previous chapters, the electronic structure of atoms has been developed by first considering the orbitals, energy levels, and states of the hydrogen atom. Subsequently, these concepts were generalized and extended to deduce similar descriptions of the properties of multielectron atoms. Because electrons occupy-ing completely filled shells of a given principal quantum number are localized around their respective nuclei, it was possible to discuss a variety of core electron spectroscopies in terms of atomic orbitals and the states derived from these. In considering the electronic structure of polyatomic molecules, it is tempting to try a rigorous quantum mechanical approach, beginning with the Schro ¨dinger equation: ^ H c i ¼ E i c i (1) where E i is the energy associated with the wave function c i . As discussed in the previous chapter, the solution to the Schro ¨dinger equation requires a specification of the form of the Hamiltonian operator: ^ H ¼ ^ T þ ^ V (2) where T ˆ and V ˆ are the respective operators for the kinetic and potential energy terms. Although one may write the equations that describe all of the kinetic 91 energy terms in a polyatomic molecule, great difficulty is encountered once equations for the potential energy terms are written. In the case of the hydrogen atom, the potential term for the electron was given by the attractive force between the negatively charged electron and the positively charged nucleus, that is, Coulomb’s Law: V ¼ e 2 = r (3) where r is the distance of the electron from the nucleus, and e is the electron fun-damental charge. For the hydrogen atom, the Hamiltonian operator took the form: ^ H ¼ h 2 2 m e r 2 e 2 r (4) where m e is the mass of the electron.
eBook - PDFCondensed Matter Optical Spectroscopy
An Illustrated Introduction
- Iulian Ionita(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
From Chapter 1, we know that atomic orbitals are described by wave-functions. Similarly, a molecular orbital is described by a molecular wavefunction. In this chapter, we will present how atomic orbitals combine to form molecular orbitals, and symmetry considerations will help show us how. For interested readers, a large collection of 218 3.1 Molecular Orbitals three-dimensional molecular orbital drawings is presented in the book by Jorgensen and Salem (1973). The following are some basic guidelines for applying of molecular orbitals theory: 1. Start with atomic orbitals. Molecular orbitals will be con-structed as linear combinations of atomic orbitals (LCAOs) from all atoms: ◾ The total number of molecular orbitals equals the total num-ber of atomic orbitals. ◾ Atomic orbitals must have a suitable symmetry. ◾ Atomic orbitals will interact if they overlap. ◾ Atomic orbitals will interact if they have closer energies. 2. Arrange the molecular orbitals (as single-electron occupancy) in order of increasing energy using the quantum mechanics protocols. 3. Add electrons to the molecular orbitals: ◾ First, fill the lowest-energy orbital. ◾ Follow the Pauli exclusion principle: add only two electrons in the same molecular orbital with their spins paired. ◾ Follow Hund’s rule: fill a degenerate molecular orbital with spin parallel electrons before pairing them. 4. Construct the molecular states from the many electron con-figurations just obtained by adding electrons to the molecular orbitals. The types of molecular orbitals are as follows: ◾ σ , with cylindrical symmetry with respect to the interatomic line, which means that the maximum electron density is located along the line; it appears as a result of the overlap between atomic orbitals oriented through the line ( s or p ).
eBook - PDFPhysical Chemistry
A Modern Introduction, Second Edition
- William M. Davis(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
322 Physical Chemistry: A Modern Introduction The correlation of atomic orbitals with molecular orbitals, as in Figure 10�7, ranks the molecularorbitalsintermsoftheirexpectedenergies�Thelowestenergyelectronicstate ofamoleculeisexpectedtoarisewhentheelectronsfillupthemolecularorbitalsfrom thebottomup,thatis,whentheyoccupythemoststableorbitals�Achemicalbondissaid toexistwhentwoelectronsoccupyabondingorbital,andagain,abondingorbitalisone withoutanodalplanebetweenthetwoatoms�H 2 hasonebond�Whenantibondingorbitals areoccupied,thenetbondingistakentobegivenbyanumbercalledthebondorder(BO): Bond order no.
- Satoru Sugano(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
10.2 Simple Description of MO Theory 251 10.2 Simple Description of MO Theory 10.2.1 MOLECULAR ORBITALS Molecular orbitals (MO) are one-electron orbitals in molecules. In the MO theory a single electron is considered to move in an average field of the nuclei and the electrons, and its motion is described by an MO. The theoretical foundation of this picture is provided by the self-consistent field (SCF) theory of Hartree or Hartree-Fock, which we will discuss in the next subsection. As seen from the argument given in Chapter I, the MO's should be the bases of the irreducible representations of the symmetry group to which the system belongs. Apart from the symmetry property, the intuitively obvious characteristic of MO is derived from the consideration that when an electron is near one nucleus, the forces exerted on it are those chiefly from the nucleus and the other electrons near it. In other words, the most important terms of the Hamiltonian for the electron near nucleus A are those which comprise the Hamiltonian for an electron in an isolated atom A. Such a consideration leads us to the approximate method of expressing MO by a suitable linear combination of atomic orbitals (AO), called the LCAO (Linear Combination of Atomic Orbitals) method. For example, we consider a molecule consisting of two atoms A and Β with atomic orbitals φ Α and φ Β , respectively. The LCAO MO in this case is Φ = C A< P A + C J W B y (10.1) in which C A and C B are numerical coefficients. For simplicity we assume that φ has no orbital degeneracy, and that the molecule has two electrons accommodated in φ, i.e., it has a closed-shell configuration. In this case we will show in the next subsection that the φ with the lowest orbital energy satisfies the equation Η = €0, (10.2) where h is an appropriate Hamiltonian for a single electron and e is the orbital energy. In general, the occupied molecular orbitals of the system satisfy Eq.
eBook - ePub- M. S. Prasada Rao(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
Chapter 7 Bonding in Molecules7.1 Molecular Orbital Theory
LCAO Method – H2 + - ionAtom is a centrosymmetric system. However, in molecules it will be disturbed. We shall make a reasonable approximation that a molecular orbital (MO) is a Linear Combination of Atomic Orbitals (LCAO).As individual hydrogen atoms at quite large distance from each other, (theoretically) are brought closer and closer, the nucleus of each atom will start to attract the electrons originally associated with the other atom. The change inenergy of the system, as a function of distance, is usually shown in the form of a curve called a Morse curve.When the distance separating the nuclei is at or near the bonding distance, the electron in the system is associated with the two nuclei instead of the original atomic orbitals on each atom. The electron is now associated with a molecular orbital(MO) that is the combination of the two atomic orbitals.When the electron is near one nucleus, the MO may be assumed to resemble the atomic orbital of the atom. Let the wave function be 01 . Similarly when the electron is in the neighborhood of the other nucleus the MO resembles the atomic orbital of the other atom. The wave function for this is given by 02 . Since the complete MO has the characteristics separately possessed by ϕ1 and ϕ2 , the total wave function for the MO is formed by the linear combination of atomic orbitals. Linear combinations are made by simple addition or subtraction of the functions to be combined. In this case it isψ=m o lϕ 1+ϕ 2This terminology, called the Liner Combination of Atomic Orbitals or LCAO method, was first suggested by R. S. Mulliken.Fig. 7.1.1 Change of energy as a function of distance.
eBook - PDFGeneral Chemistry I as a Second Language
Mastering the Fundamental Skills
- David R. Klein(Author)
- 2015(Publication Date)
- Wiley(Publisher)
Rather, these new orbitals are centered in between two atoms (and in some cases, the orbitals can be spread out over an even larger region of a molecule). So, we call them molecular orbitals. In our analogy, atomic orbitals are the two original houses, and molecular orbitals are the mansion and the out-house. The mansion is called the bonding molecular or- bital, because the bond is formed when the two electrons lower their energy by go- ing into this molecular orbital. The out-house is called the anti-bonding molecular orbital, because the two electrons would be higher in energy (less stable) if they went into this molecular orbital. In our analogy, it might be possible to pay one of the people enough money to go live in the out-house for one night. Similarly, it is possible to give one of the electrons enough energy to “jump” from the bonding molecular orbital into the anti- bonding molecular orbital (for a brief moment). There is an entire field of science devoted to how, when, and why that happens. It is called spectroscopy, which is discussed in more detail in an organic chemistry course. Increasing Energy For now, it is important to understand the energy diagram just shown. You will see many more of these diagrams in this course, and you will need to know how to read them. That is why we spent so much time on the analogy. Now that we understand what a bond is, let’s turn our attention to important types of bonds. 6.3 ELECTRONEGATIVITY, INDUCTION, AND POLARITY In the previous section, we saw how two atoms can share electrons to form a bond. But we cannot assume that the two atoms are sharing the electrons equally. To see what we mean by this, we will need to revisit our cloud analogy. We said in the beginning of this chapter that we should not think of electrons as particles that are in orbit of some nucleus. Also, we should not think of them as waves surrounding a nucleus. Rather, they are some combination of these two mod- els.
- Zeev Burshtein(Author)
- 2022(Publication Date)
- Wiley(Publisher)
6 Electron Orbits in Molecules 6.1 Preamble Molecules are usually described as a collection of bonded atoms forming a constant structure. Fundamentally, however, it should be described as a collection of nuclei and electrons interacting by electrostatic forces, forming a stable structure, in other words a bonded solution of the time-independent Schrödinger equation. In the electrostatic approximation, the molecular Hamiltonian is given by = all nuclei u 2 2M u + all electrons i 2 2m + all nucleus pairs Z u Z v e 2 R u − R v + all electron pairs e 2 r i − r j + all electron nucleus pairs −Z u e 2 R u − r i , 6 1 where u , M u , R u , and Z u are the linear momentum, mass, position vector, and atomic number of the u nucleus, respectively, i and r i are the linear momentum and position vectors of the i electron, respectively, and m is the electronic mass. The first two sums provide the kinetic energies of the nuclei and electrons, respectively; the third sum provides the electrostatic repulsion energy among electron pairs; and the last sum provides the electrostatic attraction energy between nuclei and electrons. Obviously, an accurate solution of such Hamiltonian is far more complex than any atomic many-electron one, which by itself is very complex and requires many approximations as described in Chapter 5. Two assumptions are made as a zero-order approximation: one is that all atomic nuclei are resting at permanent positions. Under such assumption, the first sum vanishes. The third sum is therefore constant, and may be ignored in considering dynamic characteristics. The second assumption is that it is possible to classify the electrons into separate sets, such that electrons belonging to the same set are always closer to a specific nucleus. In other words, they may be permanently labeled as belonging to the same nucleus, and their interaction with electrons of other sets and other nuclei may be ignored.
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