Introduction

The Bohr Theory was a milestone; though, it had major defects. It clarified the energies of the electron conversions in hydrogen and hydrogen-like atoms, but not in any atom having more than 1 electron. Even for hydrogen, the theory couldn't explain the diverse intensities of the spectral lines.

In this chapter, a more sophisticated theory of the electronic structure of the atom will be explained. This theory, recognized as wave mechanics, doesn't only provide consequences as good as the Bohr Theory for the case of the hydrogen atom but as well can be utilized to clarify both the energies and the intensities of the spectral lines of hydrogen. In addition, approximate explanations of atoms that have more than 1 electron might be given in terms of this theory.

The branch of science that obtains into account this dual behaviour of matter is termed quantum mechanics. Quantum mechanics is a theoretical science that deals through the study of the motion of the microscopic objects that have both observable wave-like and particle-like properties.

This science was build up independently through Louis de Broglie, Werner Heisenberg and Erwin Schrödinger. 

The quantum mechanical model is depending on quantum theory that says matter as well has properties associated through waves. According to quantum theory, it is impossible to know the exact position and momentum of an electron at the similar time. This is recognized as the uncertainty principle.

The quantum mechanical model of the atom utilizes compound shapes of orbitals (sometimes termed electron clouds), volumes of space in that there is likely to be an electron. So, this model is based on probability rather than certainty.

The main difference is that in the Bohr model the electrons rotate around the nucleus in fixed orbits alike to the way planets orbit around the sun. The wave mechanical model, influenced through the Heisenberg uncertainty principle, says that electrons don't orbit in fixed orbits. In reality, it is impossible to know both the position and momentum of a particle like an electron. Instead, the wave mechanical model utilizes the Schrödinger equation to calculate the probabilities of where the electron might be positioned at any following time (with no any certainty where the electron actually is). 

There are other dissimilarities (differences in shapes of electron orbits/atomic orbitals; failure of the Bohr model to account for hybrid orbitals, molecular orbitals, resonance, etc.), but the major difference is that Bohr's model provides the electron orbit an accurate travel path while the wave mechanical model doesn't maintain to know where the electron is at any following time, only probabilities of where it is likely to be).

Energy is Quantized (The Planck-Einstein Relation: E = h)

After Max Planck computed that energy is released and absorbed through atoms in certain fixed amounts recognized as quanta, Albert Einstein took his work a step additional. He stated that radiant energy is as well quantized-he termed the discrete energy packets, photons. Einstein's theory states that electromagnetic radiation (light, for example) has characteristics of both a wave and a stream of particles. 

The Planck-Einstein relation is the equation relating energy to frequency. It was the 1^st equation of quantum mechanics, implying that energy approaches in multiples ('quanta') of a fundamental steady h. It is written as:

E = hv  

v is linear frequency and w is angular frequency. The fundamental steady h is termed Planck's constant and is equal to 6.62608 ×10-34 Js 

This relation was 1^st proposed through Planck in 1900 to describe the properties of black body radiation. The interpretation was that matter energy levels are quantized. At that time, this appeared compatible through the notion that matter is composed of particles that oscillate. The finding that the energy of electrons in atoms is following through discrete levels as well fitted well by the Planck's relation.

In the year 1905, Einstein proposed that the similar equation should hold as well for photons, in his clarification of the photoelectric effect. The light incident on a metal plate provides rise to a current of electrons only whenever the frequency of the light is greater than a certain value. This value is associated through the energy needed to eliminate an electron from the metal (the 'work function'). The electron is expelled only when the light energy matches the discrete electron binding energy. Einstein's proposal that the light energy is quantized just like the electron energy was more radical at the time: light quantization was harder for people to admit than quantization of energy levels of matter particles. (The term 'photon' for such quantized packets of light energy came later, given through G. N. Lewis, of Lewis Hall!).

Wave-particle duality

In the year 1924, a young French scientist, Louis de Broglie, proposed that, like light, a stream of electrons might have wave properties additionally to particle properties. He proposed that a particle these as an electron travelling at a velocity possessed a wavelength which is given through

             λ = h/mv

Where h is Planck's constant and m is the mass of the particle. In the year 1927, two Americas, C.J. Davison and L.H. Germer, calculated the wavelength of an electron beam through diffraction through a crystal in a manner alike to the diffraction of X rays. Therefore de Broglie's hypothesis was confirmed. 

De Broglie proposed this relationship as a common one. By every particle, there is a connected wave. The wavelength of the particle based on its mass and how fast it is moving.

A particle occupies a particular location, but a wave has no exact position. A wave extends over some region of space. Because of their wave properties, electrons are always spread out rather than located in one particular space. As a consequence, the position of an electron can't be precisely described. Electrons are said to be delocalized since their waves are increase out rather than pinpointed. 

Visioning the electron and other particle-waves as delocalized, though, modifies the way we look at things. Instead of things having an accurate location and motion, they are allocated over several volumes. An electrons beam is deflected through a magnetic field; this is evidence that electrons have a particle-like nature. Electron diffraction is an evidence for the wave nature of the electron.

The uncertainty principle

This is as well recognized as the Indeterminacy principle. Werner Heisenberg, a German Physicist, found in the year 1920s that the position and motion of a particle-wave can't be "pinned down". If a particular particle-wave can be pinpointed in a specific location, its motion must be unknown.

Equally, if the motion of a particular particle-wave is well-known accurately, its location must be unknown. Heisenberg summarized this uncertainty in what has happen to known as the uncertainty principle: The more precisely position is recognized, the less precisely can the momentum be computed, and vice-versa. Mathematically we explain the uncertainty principle as the following, where 'x' is position and 'p' is momentum: 

            Δx. Δp > h/2 π

Where h is Planck's constant. The principle, that was 1^st stated through Heisenberg, arises from the dual particle-wave nature of matter.

For instance, we can compute the location of an electron, but not its momentum (energy) at the similar time.

Fig: The Uncertainty Principle

This is perhaps the most famous equation next to E = mc2 in physics. It essentially says that the amalgamation of the error in position times the error in momentum must always be greater than Planck's constant. So, we can determine the position of an electron to some accuracy, but then its momentum will be inside a very large range of values. Likewise, we can compute the momentum exactly, but then its position is unknown.

The Schrödinger equation 

In the year 1926, Erwin Schrödinger postulated an equation alike to those which explain wave motion, the solutions of that describe the properties of the electron in the atom. The solutions to Schrödinger equation are precise for the electron in the hydrogen atom and estimated for the electrons in other atoms. Such solutions are related to the probability of discovering the electron in a given locality in the atom.

The wave function

Since an electron has wave properties, it is explained as a wave function, Ψ, or Ψ(x,y,z); the latter meaning that Ψ is a function of coordinates x, y, and z. The probability of discovering an electron in any volume in space is proportional to the square of the complete value of the wave function, integrated over that volume of space. This is the physical significance of the wave function. The quantities we create of electronic charge density, then, should be related to |Ψ|², and not |Ψ|.

Expressed as an equation, we have:

Probability (x,y,z) α|Ψ(x,y,z)|2.

The Schrödinger equation is generally written in the form: HΨ = EΨ

This looks deceivingly easy.

H is termed the Hamiltonian operator and encloses expressions for the kinetic and potential energies of the system.

E is the numerical value of the energy for any particular Ψ.

The complete Schrödinger equation is:

Before the Schrödinger equation can be resolved, the exact form of V must be particular. V encloses information about all the Columbic attractions and repulsions among the electrons and nuclei of the atom of molecule.  

V = Potential energy.

The solutions of the Schrödinger equation include integers that find out the energies and momenta of the electrons. The integers correspond to the quantum numbers of the Bohr Theory, but in this case they are needed through the mathematical form of the wave equation, whereas the quantum numbers of the Bohr Theory were supposed arbitrarily. At this point, it isn't essential to learn how to answer the Schrödinger equation. It is only needed to know that the solutions are utilized to explain the arrangement of electron in atoms and that the quantum numbers are the most characteristic feature of the solutions. The arrangements of the electrons in atoms can be predicted in terms of the quantum numbers alone. The properties of electrons which are determined by this arrangement include their energies, their orientations in space, and their interactions with other electrons within the same atom and with electrons of other atoms. All the chemical and physical properties of an atom depend on the arrangement of its electrons;

Therefore the quantum numbers are instantly helpful in providing chemical information.

Each electron in an atom is assigned four quantum numbers, which define its energy, its orientation in space, and its possible interaction with other electrons.

Quantum numbers

By solving the Schrödinger equation (HΨ = EΨ), we obtain a set of mathematical equations, called wave functions  (Ψ), that explain the probability of finding electrons at definite energy levels within an atom.

A wave function for an electron in an atom is termed an atomic orbital; this atomic orbital explains a region of space in that there is a high probability of discovering the electron. Energy transforms inside an atom are the consequence of an electron changing from a wave pattern through one energy to a wave pattern through a different energy (generally accompanied through the absorption or emission of a photon of light).

Each electron in an atom is described by four different quantum numbers. The first three (n, l, ml) identify the particular orbital of interest, and the fourth (ms) specifies how many electrons can occupy that orbital.

i. Principal  Quantum Number (n):   n  = 1, 2, 3...  ∞ Specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of  n  are said to be in the same  shell  (level). For a hydrogen atom,  the most stable state is when n = 1, and levels for n > 1 are excited states of the atom.  The total number of orbitals for a given n value is n2.

ii. Angular Momentum (Azimunthal) Quantum Number (l):  l = 0, ..., n-1 As the name implies, it specifies the angular momentum of the electron. In any atom other than the hydrogen atom, n and l together classify the energy of the electron. l as well specifies the shape of an orbital through a particular principal quantum number.

The secondary quantum number separates the shells into smaller groups of orbitals termed subshells (sublevels). Generally, a letter code is utilized to identify l to avoid confusion with n:

 l            0     1      2     3     4      5    .......

Letter    s      p     d      f     g      h    .......

The subshell by n = 2 and l = 1 is the 2p subshell; if n = 3 and l = 0, it is the 3s subshell, and so on. The value of l as well has a slight result on the energy of the subshell; the energy of the subshell amplifies through l (s < p < d < f). 

iii. Magnetic Quantum Number (ml):  ml = -l, ..., 0, ..., +l   Specifies the orientation in space of an orbital of a following energy  (n) and shape (l). It describes the possible orientations of the angular momentum in space through respect to several arbitrarily described axis. The magnetic quantum number becomes significant in situations in that the electron interacts through external magnetic fields, including the fields generated via the motions of other electrons.  This number splits the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Therefore, the s subshell has only one orbital, the p subshell has 3 orbitals, and so on.

iv. Spin Quantum Number (m_s):  m_s = +½ or -½. To account for the fine features examined in atomic spectra, it is needed to initiate the 4^th quantum number; ms that specifies the orientation of the spin axis of an electron. This quantum number obtains into account the spinning of the electron about its own axis as it moves about the nucleus. An electron can spin in only one of 2 directions. Even though ms has the possible non-integral values -½ and +½, the difference between the permitted values is integral. 

Table: Allowed Quantum Numbers

Table:  Sublevels are broken down into orbitals;  each  orbital holds 2 electrons

Orbitals

Atomic orbital: This is the volume enclosing all the points within a free atom at which the wave function of an electron has a considerable magnitude. More basically, orbital is a region around the nucleus of an atom where the probability of discovering an electron is high. 

A solution of the Schrödinger equation, expressed in terms of a set of permitted values of n, l, and m_l, describes an orbital. The most characteristic property of an orbital is its energy. Since in the absence of outside field the value of ml doesn't influence the energy, orbitals are grouped into sets termed subshells that are denoted through the values of n and l only. The employ of 2 numbers is awkward, therefore orbitals through l values of 0, 1, 2, and 3, are signified as s, p, d, and f, correspondingly.

The energies of atomic orbitals as well explain their shapes. The shapes are doubtful, but forecasts have been made through experimentation. Another hard task is describing where an electron is. We can imagine of it as a wave, and describing its exact location is not possible for us to understand. Instead, we can think of it as the statistical probability of the electron being originate at a particular place. At any particular moment, an electron can be established at any distance from the nucleus and in any direction according to the Heisenberg Uncertainty Principle. 

The s-Orbital

The s orbital is a spherically-shaped region describing where an electron can be found, within a certain degree of probability. The shape of the orbital depends on the quantum numbers associated with an energy state. All s orbitals have l = m = 0, but the value of n can vary.

Fig: The s-Orbitals

The p-Orbital

The p-orbital is a dumbbell-shaped region describing where an electron can be originated, within a definite degree of probability. The shape of the orbital depends on the quantum numbers connected through an energy state. Since there are 3 p-orbitals per energy level, the lobes expand out along the x-axis (px orbital), the y-axis (py orbital), and the z-axis (pz orbital). 

Fig: The p-Orbitals

The d-Orbital

The d orbital's shapes are even more compound since there are 5 orbitals in a d subshell. Four of the five d-orbitals (dxy, dxz, dyz, and dx₂-y₂) have 4 lobes expanding out perpendicular to each other. The last one, dz2, has two lobes extending out along the z-axis with a torus (doughnut-shaped ring) around the centre on the x-y plane. 

Fig: The d-Orbitals

The energy of an electron in a given orbital can be particular accurately from the values of its quantum numbers. As a consequence, the position of the electron is quite uncertain. Consequently, it turns out that whilst the solutions to the Schrödinger equation give the energies of the electron, they can provide only the probability of finding the electron in any arbitrary volume around the nucleus.

The probability distribution for an electron in a 1s, 2s or 3s orbital of hydrogen is:

Fig: Radial Distribution Functions of the 1s and 2s Orbitals

As we know that as the principal quantum number rises, the maximum in the probability density occurs at larger distances for orbitals of the similar angular momentum type. For instance, the maximum probability for the 1s orbital take place at 1 bohr (0.529 Å), for the 2s orbital the primary maximum take places at 5.3 bohr, and for the 3s orbital the primary maximum happens at about 13 bohr. 

For the hydrogen atom, it is seen that there is maximum probability of discovering the electron at a distance of 1 bohr (0.529 Å) from the nucleus. This distance is the similar as the experimental radius of the hydrogen atom and as well concurs through the radius derived through means of the Bohr Theory. 

Alike illustration for the 2s orbital of hydrogen is shown - there is a greater probability of finding the electron additional away from the nucleus than in the case of the 1s orbital, but there is still several possibility of discovering the electron extremely close to the nucleus. As well important is the fact that at an intermediate distance, there is a surface at that the probability of finding the electron is zero. Surfaces at which the probability goes to zero are termed nodes. For any orbital having principal quantum number n, there are always n-1 nodes. This nodal character of the probability distribution is consistent through the assumption that the motion of the electron has the character of a wave.

Writing electron configuration 

The Pauli Exclusion Principle

The Pauli Exclusion Principle (Wolfgang Pauli, Nobel Prize 1945) states that no two electrons in the same atom can have identical values for all four of their quantum numbers. What this means is that no more than two electrons can occupy the same orbital, and that two electrons in the same orbital must have opposite spins.

Since an electron spins, it generates a magnetic field, which can be oriented in one of the two directions. For 2 electrons in the similar orbital, the spins must be conflicting to each other; the spins are said to be paired. Such substances aren't attracted to magnets and are said to be diamagnetic. Atoms through more electrons that spin in one direction than another enclose unpaired electrons. Such materials are weakly attracted to magnets and are said to be paramagnetic.

The aufbau principle

The distribution of electrons among the orbitals of an atom is termed the electron configuration. The electrons are packed in according to a scheme identified as the Aufbau principle ("building-up"), that corresponds (for the most part) to rising energy of the subshells:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f

It isn't needed to memories this listing, because the order in that the electrons are filled in can be read from the periodic table in the following fashion:

Fig: Writing Electronic Configurations

In electron configurations, write the orbitals that are occupied via electrons, followed through a superscript to point to how many electrons are in the set of orbitals (for instance H 1s¹).

Another way to indicate the placement of electrons is an orbital diagram, in that each orbital is symbolized through a square (or circle), and the electrons as arrows pointing up or down (indicating the electron spin). 

The Hund's Rule of Maximum Multiplicity

When electrons are positioned in a set of orbitals of equal energy, they are increase out as much as feasible to provide as few paired electrons as possible (Hund's rule).

According to this rule, electron pairing will not occur in orbitals of similar energy (similar sub-shell) until each orbital is 1^st singly filled through parallel spin. In other terms, in a set of orbitals having similar energy (degenerate orbitals), the electrons distribute themselves to occupy divide orbitals through similar spin as far as possible. This rule can be demonstrated through considering the instance of carbon. The atomic number of carbon is 6 and it includes 2 electrons in 2p subshell and such can be distributed in the following 3 ways: 

Fig: Hund's Rule of Maximum Multiplicity

Because all the three 2p orbitals have similar energy, hence, it doesn't acquire any difference as to which of the 3 orbitals enclose electrons. In state (a), both the electrons are in the similar orbital. In condition (b), the 2 electrons are present in different orbitals but through opposite spins while in state (c), the electrons are present in different orbitals through similar spins. Now, the electrons are charged particles and repel one another.

The electron-electron repulsions are minimum whenever the electrons are as far apart as possible through parallel spins. Therefore, state (c) has minimum repulsions and corresponds to lower energy (stable) state. This is in accordance by Hund's rule. This principle is extremely significant in guiding the filling of p, d and f subshell that have more than one kind of orbitals.

In a ground state configuration, all of the electrons are in as low an energy level as it is possible. Whenever an electron absorbs energy, it occupies a higher energy orbital, and is said to be in an excited state.    

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