Number, quantum

Quantum numbers specify the address of each electron in an atom. There are four types of quantum numbers  

There are no two electrons in an atom that can have the same four quantum numbers. Each electron has a unique address, like a family living in a flat. This is Pauli s Exclusion Principle. 

Let us consider the address of a family. Imagine that a family migrates and seeks a new flat in another district. In that district there are K, L, M, N...streets which contain s, p, d, f... apartments with one, three, five and seven floors respectively and two flats in each floor. That family is not a rich one and there are many kinds of flats with varying rents. When a street or apartment is changed the rent to be paid is changed too. Flats in K street have the lowest rents, and then L, etc. The family could decide, for example, to live in L street, p apartment, py floor and clockwise flat. 

Schematic representation of quantum numbers and electron distribution. 

The table below shows the quantum numbers of electrons in the first four shells. 

Principal quantum number Angular momentum quantum number n Any positive integer 

The angular momenrnm quanmm number is denoted /. It also affects the energy of the electron, but in general not as much as the principal quanmm number does. In the absence of an electric or magnetic field around the atom, only n and / have any effect on the energy of the electron. The value of / can be 0 or any positive integer up to, but not including, the value of n for that electron. 

The magnetic quantum number, denoted mi, determines me orientation in space of me electron, but does not ordinarily affect me energy of me electron. Its values depend on the value of / for mat electron, ranging 

The spin quantum number, denoted m, is related to the spin of the electron on its axis. It ordinarily does not affect the energy of the electron. Its possible values are — and +. The value of does not depend on the value of any other quantum number. 

The permitted values for the other quantum numbers when n = 2 are shown in Table 4-2. The following examples will illustrate the limitations on the values of the quantum numbers (Table 4-1). 

A set of four quantum numbers is the address or ID number for an electron in a given atom. No two electrons in the same atom can have the same four quantum numbers. 

The first quantum number is the principal quantum number, n. The principal quantum number designates the shell level. The larger the principal quantum number, the greater tire size and energy of the electron orbital. For the representative elements the principal quantum number for electrons in the outer most shell is given by the period in the periodic table. The principal quantum number for tire transition metals lags one shell behind the period, and for the lanthanides and actinides lags two shells behind tire period. 

Valence electrons, the electrons which contribute most to an element s chemical properties, are located in the outermost shell of an atom. Typically, but not always, only electrons from the s and p subshells are considered valence electrons. 

The second quantum number is the azimuthal quantum number, t The azimuthal quantum number designates the subshell. These are the orbital shapes with which we are familiar such as s, p, d, and f. If C = 0, we are in the s subshell if = 1, we are in the p subshell and so on. For each new shell, there exists an additional subshell with the azimuthal quantum number f = it -1. Each subshell has a peculiar shape to its orbitals. The shapes are based on probability functions of the position of the electron. There is a 90% chance of finding the electron somewhere inside a given shape. You should recognize the shapes of the orbitals in the s and p subshells. 

The third quantum number is the magnetic quantum number, mr The magnetic quantum number designates the precise orbital of a given subshell. Each subshell will have orbitals with magnetic quantum numbers from -1 to +fi. Thus for the first shell with n - 1, and 1 = 0, there is only one possible orbital, and its magnetic quantum number is 0. For the third shell with n- 3, and C = 2, there are 5 possible orbitals which have the magnetic quantum numbers of -2, -1, 0, +1, and +2. 

In quantum mechanics, three quantum numbers are required to describe the distribution of electrons in hydrogen and other atoms. These numbeis arc derived from the mathematical solution of the Schrodinger equation for the hydrogen atom. They are called the principal quantum number, the angular momentum quantum number, and the magnetic quantum number. These quantum numbeis will be used to describe atomic oibitals and to label electrons that reside in them. A fourth quantum number— the spin quantum number—describes the behavior of a specific electron and completes the description of electrons in atoms. 

A collection of orbitals with the same value of n is frequently called a shell. One or more orbitals with the same n and values are referred to as a subshell. For example, the shell with n = 2 is composed of two subshells, = 0 and 1 (the allowed values for n = 2). These subshells are called the 2s and Ip subshells where 2 denotes the value of n, and s and p denote the values of . 

The magnetic quantum number mf) describes the orientation of the orbital in space (to be discussed in Section 7.7). Within a subshell, the value of m depends on the value of the angular momentum quantum number, . For a certain value of , there are (2F + 1) integral values of m as follows  

The (a) clockwise and (b) counterclockwise spins of an electron. The magnetic fields generated by these two spinning motions are analogous to those from the two magnets. The upward and downward arrows are used to denote the direction of spin. 

Orbital Shapes and Energy Online Learning Center, Interactives 

Click Coached Problems for a self-study module on quantum ntimbers. 

This relation between n and comes from the Schrodinger equation. 

The Schrodinger equation can be solved approximately for atoms with two or more electrons. There are many solutions for the wave function, ij/, each associated with a set of numbers called quantum numbers. Three such numbers are given the symbols n, , and mi. A wave function corresponding to a particular set of three quantum numbers (e.g., n = 2, = 1, mi = 0) is associated with an electron occupying an atomic orbital. From the expression for ij/y we can deduce the relative energy of that orbital, its shape, and its orientation in space. 

For reasons we will discuss later, a fourth quantum number is required to completely describe a specific electron in a multielectron atom. The fourth quantum number is given the symbol ms. Each electron in an atom has a set of four quantum numbers n, l, mi, and ms. We will now discuss the quantum numbers of electrons as they are used in atoms beyond hydrogen. 

First Quantum Number, n Principal Energy Levels 

As spin-orbit coupling is weak in conjugated systems the total spin is a conserved quantum number. The low-lying energy eigenstates are singlet S = 0) and triplet (5=1) states. 

Tie state of an electron in an atom can be completely described by four quantum numbers, designated as n,. mf, and ms. The first, or principal, quantum number, n, indicates the electron s approximate distance from the nucleus. The second quantum number, describes the shape of the electron s orbit around the nucleus. The third quantum number, mf, describes the orientation of the electron s orbit compared to the plane of the atom. The fourth quantum number, ms, tells the direction of the electron s spin (clockwise or counterclockwise). 

The Schrodinger wave equation imposes certain mathematical restrictions on the quantum numbers. They are as follows  

As an example, consider electrons in the first energy level of an atom, that is, n = 1. In this case, can have any integral value from 0 to (n — 1), or 0 to (1 - 1). In other words, 

Use the rules given above to complete the table listing the quantum numbers for each electron in a boron atom. The correct quantum numbers for one electron in the atom is provided as an example. 

The description of the electronic structure of an atom begins with four numbers called quantum numbers. The four quantum numbers are described as follows  

I mean higher values of angular momentum. There are letter designations associated with values of I, just as there are letter designations associated with values of n  

Because there are n values of / for a given value of n, this means there are n subshells in a given shell. Here is an example of how we can list shells and subshells as follows  

The values of m, designate orbitals, regions of space occupied by electrons with shapes that are determined by the wave properties possessed by the electrons. (The term orbital is derived from the term orbit. Orbits, however are only two-dimensional. The term orbital is used to emphasize that it is referring to something that is three-dimensional.) Thus, in a subshell designated by the quantum number /, there are 21 + 1 orbitals. Here are some examples  

Each electron in an atom possesses a set of four quantum numbers, designated n, /, m, and m. Two electrons cannot possess the same set of quantum numbers at the same time. An electron can, however, change the values of its quantum numbers by making transitions to different energy levels. 

Because only a fixed number of electrons can exist in each stationary state (Pauli exclusion principle), each bound electron has its own set of quantum numbers (these are discussed in more detail in Section 2.2.1.1). Each stationary state is also described by specific notation (spectroscopic ox X-ray notation as discussed further in Section 2.2.1.2) irrespective of whether they contain electrons or not. 

1 Quantum Numbers Quantum numbers describe a scheme based on quantum mechanics in which the energy, momenta, spatial distribution, and spin of each bound electron is specified. 

An electron for which n = 1 is said to be in the first principal level. If n = 2, we are dealing with the second principal level, and so on. 

Each principal energy level includes one or more sublevels. The sublevels are denoted by the second quantum number, . As we will see later, the general shape of the electron cloud associated with an electron is determined by . Larger values of produce more complex shapes. The quantum numbers n and are related can take on any integral value starting with 0 and going up to a maximum of (n 1). That is. 

Although we don t need detailed treatments of quantum mechanics to understand light bulbs or most introductory chemical concepts, we will find that the vocabulary established by the theory is important. For example, the names attached to atomic orbitals come from the functions that solve the wave equations. Collectively, they are referred to as quantum numbers. 

Other waves are also possible without moving the ends of the string. If a guitarist wants to play what is called the harmonic, a finger is placed very lighdy at the midpoint of the string. The string can still vibrate on both sides of the 

Suppose that we wanted to write a general equation for the waves that can be formed in our string. Working from the cases above, you should be able to see that the necessary form will be 

When we solve the Schrodinger equation for an atom, the resulting wave functions are much more complicated than these sine waves. To write the mathe- 

The principal quantum number defines the shell in which a particular orbital is found and must be a positive integer ( = 1, 2, 3, 4, 5,.. . ). When n= 1, we are describing the first shell when n = 2, the second shell and so on. Because there is only one electron in hydrogen, all orbitals in the same shell have the same energy, as predicted by the Bohr model. However, when atoms have more than one electron, the negative charges on the electrons repel one another. This repulsion between electrons causes energy differences between orbitals in a shell. This difficulty was one reason the Bohr model had to be replaced. 

The Schrodinger equation specifies the possible energy states the electron can occupy in a hydrogen atom and identifies the corresponding wave functions (if/). These energy states and wave functions are characterized by a set of quantum numbers (to be discussed shortly), with which we can construct a comprehensive model of the 

To distinguish the quantum mechanical desaiplion of an atom from Bohr s model, we speak of an atomic orbital, rather than an orbit. An atomic orbital can be thought of as the wave function of an electron in an atom. When we say that an electron is in a certain orbital, we mean that the distribution of the electron density or the probability of locating the electron in space is described by the square of the wave function associated with that orbital. An atomic orbital, therefore, has a characteristic energy, as well as a characterislic distribution of electron density. 

In order to begin to understand the behavior of atoms, we must first look at some of the details of the quantum mechanical model of the atom. Schrodinger s equation predicts the presence of certain regions in the atom where electrons are likely to be found. These regions, known as orbitals, are located at various distances from the nucleus, are oriented in certain directions, and have certain characteristic shapes. Let s look at some of the basic components of the atom as predicted by the equation, and at the same time we will review quantum numbers. 

If you have four pieces of information about a house, you can find it just about anywhere in the United States. These four pieces are the Street Number, the Street Name, the City, and the State. So it is with atoms. There are four pieces of information from which you can identify a specific electron within any atom. These are known as quantum numbers, a list of which is shown below. 

Principal quantum number (n)—This number describes the energy level in which the electron can be found. In our house analogy, this is equivalent to the State, which is the most general information about where the house is located. These correspond to regions that are found at specific distances from the nucleus of the atom. The principal quantum number, n, has a whole number, positive value, from 1 to 7 (n = 1,2, 37). Lower values of n correspond to orbitals close to the nucleus and lower energy levels. 

Azimuthal quantum number (1)—This number describes the shape of the orbital. The azimuthal quantum number can have values, from 0 to n-1, and these values correspond to certain orbital shapes. While the value can theoretically have a value as high as 6, we will see later that no values higher than 3 are found. The values that do exist, 0, 1, 2, and 3, correspond to particular shapes and are commonly designated as s, p, d, and /orbitals, respectively. In our house analogy, this quantum number would correspond roughly to the City. That is, it is a bit more specific than the State, but it still doesn t tell us exactly where the house is. 

Magnetic quantum number (m or —This number describes the orientation of the orbital 

Magnetic quantum number (m or znj)—This number describes the orientation of the orbital in space. The value of w is a range from -1 to +1, including 0, In the house analogy, this quantum number is similar to the street name of the house. You can tell about where the house is from this information, but not exactly. In much the same way, you can determine which orbital an electron should be in from the magnetic quantum number, but you can t identify the specific electron. 

It was the analysis of the line spectrum of hydrogen observed by J. J. Balmer and others that led Neils Bohr to a treatment of the hydrogen atom that is now referred to as the Bohr model. In that model, there are supposedly allowed orbits in which the electron can move around the nucleus without radiating electromagnetic energy. The orbits are those for which the angular momentum, mvr, can have only certain values (they are referred to as quantized). This condition can be represented by the relationship 

In 1924, Louis de Broglie, as a young doctoral student, investigated some of the consequences of relativity theory. It was known that for electromagnetic radiation, the energy, E, is expressed by the Planck relationship, 

A specific photon can have only one energy, so the right-hand sides of Eqs. (2.2) and (2.3) must be equal. Therefore, 

The product of mass and velocity equals momentum, so the wavelength of a photon, represented by h/mc, is Planck s constant divided by its momentum. Because particles have many of the characteristics of photons, de Broglie reasoned that for a particle moving at a velocity, v, there should be an associated wavelength that is expressed as 

This predicted wave character was verified in 1927 by C. J. Davisson and L. H. Germer who studied the diffraction of an electron beam that was directed at a nickel crystal. Diffraction is a characteristic of waves, so it was demonstrated that moving electrons have a wave character. 

Thus the eigenvalues of lz can alter only as integer multiples ofH so that (i — nh.fi —fi, At, At +/ .fi + nK are allowed. This series has its lowest member, Atmjn, and the highest member, Atmax restricted by the shift operators 

1 A = 10 m the angstrom is most often used as the unit for atomic radius because of its convenient size. Another convenient unit is the picometer  

One more characteristic of the hydrogen I5 orbitai that we must consider is its size. As we can see from Fig. 7.12, the size of this orbitai cannot be defined precisely, since the probability never becomes zero (although it drops to an extremely small value at large values of r). So, in fact, the hydrogen I5 orbital has no distinct size. Flowever, it is useful to have a definition of relative orbital size. The definition most often used by chemists to describe the size of the hydrogen Is orbital is the radius of the sphere that encloses 90% of the total electron probability. That is, 90% of the time the electron is inside this sphere. 

So far we have described only the lowest-energy wave function in the hydrogen atom, the I5 orbital. Flydrogen has many other orbitals, which we will describe in the next section. Flowever, before we proceed, we should summarize what we have said about the meaning of an atomic orbital. An orbital is difficult to define precisely at an introductory level. Technically, an orbital is a wave function. Flowever, it is usually most helpful to picture an orbital as a three-dimensional electron density map. That is, an electron in a particular atomic orbital is assumed to exhibit the electron probability indicated by the orbital map. 

When we solve the Schrodinger equation for the hydrogen atom, we find many wave functions (orbitals) that satisfy it. Each of these orbitals is characterized by a series of numbers called quantum numbers, which describe various properties of the orbital  

The angular momentum quantum number ( ) has integral values from 0 to - 1 for each value of . This quantum number is related to the shape of atomic orbitals. The value of for a particular orbital is commonly assigned a letter = 0 is called s  

A photon is emitted when an electron in an atom jumps from a higher to a lower energy level. The energy of the emitted photon is equal to the difference in energy between the two energy levels. 

All electrons present in an atom have specific addresses or attributes by which each electron can be referred to. The four quantum numbers are the ones with which we can describe each and every electron that is present in an atom. One of die quantum numbers describes the sh or the most probable area around die nucleus where we can find the particular electrons of interest. This wave function of an electron is called an orbital. 

Principal quantum number ( ). The principal quantum number denotes the energy level of electrons. The larger the principal quantum number is, the larger the energy. The smaller the principal quantum number is, the lower the energy. The shells are often named K, L, M, N. which correspond to the principal quantum numbers 1,2, 3,4. respectively. 

The orbital size depends on n. This means that the larger the n value, the larger the orbital. Orbitals with the same n belong to the same shell. 

Angular momentum quantum number (/). Angular momentum quantum number (azimuthal quantum number) denotes the shape of the orbital. The values range from 0 to n - 1, where n stands for the principal quantum number. If an electron has a principal quantum number of 4, the values of angular momentum quantum numbers are 0,1,2, and 3. The angular momentum quantum numbers correspond to different subshells. An angular momentum quantum number 0 corresponds to s subshell, 1 to subshell, 2Xod subshell, 3 to/subshell, and so on. For instance, M denotes a subshell with quantum numbers = 3 and 1 = 1. 

This equation has been solved exactly only for one-electron species such as the hydrogen atom and the ions He and Li. Simplifying assumptions are necessary to solve the equation for more complex atoms and molecules. Chemists and physicists have used their intuition and ingenuity (and modern computers), however, to apply this equation to more complex systems. 

In 1928 Paul A. M. Dirac (1902-1984) reformulated electron quantum mechanics to take into account the effects of relativity. This gave rise to an additional quantum number. 

Unless otheiwise noted, all content on this page is Cengage Learning. 

Let s define each quantum number and describe the range of values it may take. 

The principal quantum number, n, describes the main energy level, or shell, that an electron occupies. It may be any positive integer  

IBLG See questions from Quantum Numbers and Orbital Shapes and Energies  

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An s orbital is spherically symmetrical and can contain a maximum of two electrons with opposed spins. A p orbital has a solid figure-of-eight shape there are three equivalent p orbitals for each principal quantum number they correspond to the three axes of rectangular coordinates. 

The d and f orbitals have more complex shapes there are five equivalent d orbitals and seven equivalent f orbitals for each principal quantum number, each orbital containing a maximum of 2 electrons with opposed spins. 

The above definitions must be qualified by stating that for principal quantum number I there are only s orbitals for principal quantum number 2 there are only s and p orbitals for principal quantum number 3 there are only s, p and d orbitals for higher principal quantum numbers there are s, p, d and f orbitals. 

Pauli exclusion principle In any atom no two electrons can have all four quantum numbers the same. See exclusion principle. 

Nuclear magnetic resctnance involves the transitions between energy levels of the fourth quantum number, the spin quantum number, and only certain nuclei whose spin is not zero can be studied by this technique. Atoms having both an even number of protons and neutrons have a zero spin for example, carbon 12, oxygen 16 and silicon 28. 

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

Figure Al.1.1. Wavefimctions for the four lowest states of the hamronie oseillator, ordered from the n = Q ground state (at the bottom) to tire u = 3 state (at the top). The vertieal displaeement of the plots is ehosen so that the loeation of the elassieal turning points are those that eoineide with the superimposed potential fimetion (dotted line). Note that the number of nodes in eaeh state eorresponds to the assoeiated quantum number.

The allowed quantum numbers are and and the corresponding eigenfimctions are usually written as a... 

V l5ini7iand S = I, respectively.. STmist be positive and can assume either integral or half-integral values, and the quantum numbers lie in the mterval... 

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

Since it is not possible to generate antisynnnetric combinations of products if the same spin orbital appears twice in each tenn, it follows that states which assign the same set of four quantum numbers twice cannot possibly satisfy the requirement P.j i = -ij/, so this statement of the exclusion principle is consistent with the more general symmetry requirement. An even more general statement of the exclusion principle, which can be regarded as an additional postulate of quantum mechanics, is... 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

However, the reader may be wondering, what is the connection of all of these classical notions—stable nonnal modes, regular motion on an invariant toms—to the quantum spectmm of a molecule observed in a spectroscopic experiment Recall that in the hannonic nonnal modes approximation, the quantum levels are defined by the set of quantum numbers (Up. . Uyy) giving the number of quanta in each of the nonnal modes. 

Wliat does one actually observe in the experunental spectrum, when the levels are characterized by the set of quantum numbers n. Mj ) for the nonnal modes The most obvious spectral observation is simply the set of energies of the levels another important observable quantity is the intensities. The latter depend very sensitively on the type of probe of the molecule used to obtain the spectmm for example, the intensities in absorption spectroscopy are in general far different from those in Raman spectroscopy. From now on we will focus on the energy levels of the spectmm, although the intensities most certainly carry much additional infonnation about the molecule, and are extremely interesting from the point of view of theoretical dynamics. 

Figure Al.2.8. Typical energy level pattern of a sequence of levels with quantum numbers nj for the number of quanta in the symmetric and antisymmetric stretch. The bend quantum number is neglected and may be taken as fixed for the sequence. The total number of quanta (n + n = 6) is the polyad number, which...

There has been a great deal of work [62, 63] investigating how one can use perturbation theory to obtain an effective Hamiltonian like tlie spectroscopic Hamiltonian, starting from a given PES. It is found that one can readily obtain an effective Hamiltonian in temis of nomial mode quantum numbers and coupling. 

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