Basic and fundamental of quantum mechanics (Theory)

Quantum Chemistry (Completely Theory)

1. Basic and Fundamental of Quantum Mechanics ( Only theory Part ) Mr.Halavath Ramesh Department of Chemistry Loyola College-Chennai University of Chemistry E-mail: [email protected]

2. Introduction Quantum Mechanics is one of the greatest intellectual developments of the 20 th century.Today,quantum mechanics has merged and assisted almost all disciplines of science such as chemistry ,physics,biology,medicine,computing and soon. Quantum Mechanics is one of the most remarkable discoveries of contemporary sciences over the last century. Quantum mechanics arose from Max Plank’s solution in 1900 to the black-body radiation and Albert Einstein’s 1905 paper on photoelectric effect. The term “ Quantum Mechanics” was first used in a paper by Max Born in 1924.However,even after 116 years of the birth of concept, it is still considered the weirdest of all sciences. The learner is not able to digest the quantum concepts and is unable to appreciate the importance of quantum mechanic in science .

3. A chemist’s intention to learn quantum mechanics is to be able to understand the structure ,Bonding and Reactivity of /in/between chemical entities be it an atom or a molecules, which is defined by the behaviour of electrons(microscopic matter).To be able to appreciate the role of quantum mechanics in chemistry, one must have a sound background in quantum mechanics. So, here is an attempt to make you understand the beauty of the subject. What is Quantum Mechanics? The term “ Quantum Mechanics” is made up of two word: QUANTUM + MECHANICS. The term “Mechanics” refers to

4. The science of the motion of the body. The other word is “Quantum” which is Latin for “ amount” and in modern conventions is used to represent the smallest possible discrete unit of any physical property. Quantum Mechanics replaces Classical Mechanics at the atomic or subatomic levels( electron and nuclei in atoms and molecules).It gives the laws of motions of microscopic objects (the way classical mechanics gives for macroscopic objects).So, we can say that Quantum Mechanics is the theoretical sciences of microscopic matter. But then what happens to matter at the microscopic level that classical mechanics? After all , matter is matter i.e., anything that has mass and occupies space.

5. Classical mechanics failed to explain certain experimental phenomena related to microscopic matter correctly which led to the origin of quantum mechanics. In fact , this difference puzzles people as there is a complete disconnect between the understanding of the universe based on classical mechanics and quantum mechanics. This is because we don’t build our understanding of quantum mechanics. Although we use identical terminology such as particles , wave , position, momentum,etc.as in classical mechanics, in quantum mechanics they attain a very different nature.

6. Origin of Quantum Mechanics Till the end of the 19 th century, Classical Mechanics was considered to be the only right and undisputed theoretical science. But soon some new experimental phenomena were observed which could not be explained by classical mechanics. This included Black Body Radiation, Photo-electric effect, Compton effect, Atomic Spectrum , Heat capacity of solid and so on. In all ,the correct interpretations of the above mentioned experimental phenomena gave two important conclusions: 1.Energy is quantized or one can say that it can be transferred only in discrete packets called quanta. 2. Light ( or radiation ) exhibits particle –like behaviour.

7. These two new concepts were the basis for the origin of the field of quantum mechanics. Classical Mechanics failed to explain the above mentioned experimental phenomena as classical mechanics considered an electromagnetic radiation solely a wave phenomenon . Now we find that there is some particles character associated with radiation ( along with the wave phenomenon ) as well. It seems that radiation shows a dual nature. In some cases, it behaves like a wave ( reflection, refraction , diffraction,etc.) and some times it manifests itself as a particle , the photon ( a photon is a single quantum of electromagnetic energy or one can say , photons are quanta of

8. Electromagnetic energy which means that the energy is quantized and can only be transferred in discrete units ( or packets) of size hv (v is the frequency) , Compton effect,etc.). Neither picture is wrong. Energy is Quantized Black body radiation Atomic spectrum Heat capacity of solids Light is composed of particles Photoelectric effect Compton effect

9. In 1924, Louis De Broglie suggested some logic to this situation. Broglie said that nature manifests itself in two forms- matter and radiation .And if radiation has dual behaviour, then by virtue of symmetry matter should also have dual behaviour. Broglie suggested that particles have wave like properties characterised by a wavelength as λ = h/p , where λ is the wavelength (wave nature) and p is the momentum of the particles [ p= mv ( v is the velocity ) particles nature]. The wave nature of matter was first confirmed by Davisson and Germer experiment. This established the wave particles Duality in matter.

10. A direct consequence of wave particles duality of matter as well as radiation led to the Heisenberg Uncertainty principle which states that the position and momentum of a particles cannot be simultaneously measured with arbitrarily high precision Δ x. Δ px >- h/4 π . Thus the more precisely we determine a particles' position ,the more we disturb its motion (momentum). With the Heisenberg uncertainty principle came the concept of “ Orbital” Bohr’s atomic model was one model that was widely appreciated but was later replaced by the quantum theory of atom. Bohr postulated that the electrons in an atom revolve round the nucleus in fixed circular paths called

11. orbits. This concept of orbit is not valid as per the Heisenberg uncertainty principle because the trajectory of a particles can only be defined if its position and momentum are known simultaneously with precision. An important consequence of the Heisenberg uncertainty principle is that one cannot determine the path of a moving microscopic particle. Bohr’s concept of orbit failed and was replaced by orbital. Here came in the concept of probability. In terms of uncertainty principle, one can only predict the probability or relative chances of locating an electron in a particular region of space around the nucleus i.e , ..one can only predict where an ele ctron is most likely to be found.

12. Quantum Mechanics replaces Classical Mechanics at the atomic or subatomic levels ( electrons and nuclei in atoms and molecules). It gives the laws of motion of microscopic objects ( the way classical mechanics gives for macroscopic objects).So, we can say that Quantum Mechanics is the theoretical science of microscopic matter.

13. Fundamental of Quantum Mechanics The dual behaviour of matter and uncertainty principle gave birth to quantum mechanics. These ideas inspired Schrodinger and Heisenberg and they independently formulated quantum mechanics in 1925 ( Schrödinger –wave mechanics and Heisenberg –Matrix mechanics) ,to study the behaviour of microscopic matter. At first sight, the two approaches appeared different but later Dirac and Newman showed that in essence the two formulations are mathematically equivalent. Here ,we will be highlighting the basis of the popular Schrodinger quantum theory only.

14. Schrodinger Quantum Mechanics: Schrodinger proposed quantum theory to explain the behaviour of microscopic particles taking into account the wave nature of particles as suggested by de broglie.This approach revolves around a partial differential equation now popularly known as the Schrödinger equation, which describes the behaviour of microscopic particles by means of a function called the wave –function There are two forms of Schrödinger equation : 1. Time dependent ( used for non-conservative systems where energy changes with time) 2. Time independent ( deals with conservative systems where energy of the system remains constant with respect to time).

15. 1. The wave –function is a function of particle’s position and time , ( x,y,z,t ) in the time dependent Schrödinger equation , whereas it is a function of position only , ( x,y,z ) in the time independent Schrödinger equation. Over here , we will be restricting our discussion to Schrödinger's time independent quantum mechanics.

16. 2. The time independent Schrodinger equation is of the form, Where the Hamiltonian operation H^ ( an operation is a mathematical command that tells you what to do and what follows; for every measurable property or observable in classical mechanics, there is a corresponding operator in quantum mechanics)acts on the wave function si and the result is proportional to the same wave function si (stationary state) and the proportionality constant, E which is the energy of the state si.H ^ is taken as sum of kinetic energy operator (T^) and potential energy operation (v^); H^=(T^) +(v^)

17. Wave-Function: wave function si (also called state function or Eigen function) is the store house of information and is the heart of Schrödinger equation as it contains all the information about the system it describes. Interestingly, wave function in itself does not have any explicit meaning. Max Born gave the correct statistical interpretation of the wave function for which he was awarded the Nobel Prize in 1954.According to born,si has no physical significance. It is merely a mathematical function of the coordinates of the system. He called si as probability amplitude and si square or si.si quare.star,the probability density of the system is the measure of probability density at that point ( probability of finding a particles in a given space).

18. In the context of atoms, an atomic orbital( si ) is a three dimensional region around the nucleus within which the probability of finding an electron with a certain energy is maximum . There is no limit to the number of solutions of the Schrödinger equation. However, a number of conditions are required for a physical realistic solution of wave function. So, an acceptable well behaved wave- function is the one which is single valued, continuous and doubly differentiable, finite, satisfy boundary conditions and normalized. Operation of Quantum Mech-anics Having studied the basics of Schrödinger quantum mechanics, the problem is how to solve a system quantum mechanically using the Schrödinger equation the following four steps are followed: Writing the Schrödinger equation for the system Defining boundary conditions Solution of Schrödinger equation Extracting information out of wave function.

19. With this four step mechanism one can obtain the entire information about the system in terms of wave function and associated energy for a given state. In fact for solving any problem quantum mechanically ,this four step process is required to be followed. However, the most difficult part of this four step process is the solution of Schrödinger equation as there is no universally accepted unique method to solve this equation. The entire quantum science revolves around the solution of this equation for a given system. Quantum Mechanics & Atomic Structure: Several attempts were made to explain the structure of an atom but the correct interpretation came in the 19 th century with quantum mechanics. The simplest chemical system-the hydrogen atom-consists of one electron and one nucleus. The most rewarding outcome of the solution of Schrödinger

20. Equation for the hydrogen atom is the occurrence of a set of integers called quantum Number, which are n( Principal quantum number; n=1,2,3….) l( Azimuthal quantum number=0,….n-1) and m(Magnetic quantum number;m—1,…,0…,+1).The quantum numbers so defined help to designate the electron present in an orbital. The distribution of electrons of an atom in its various orbital's gives the electronic configuration. So, the quantum mechanical solution of the hydrogen atom lays the foundation of the Atomic structure. Quantum Mechanics & Spectroscopy : Quantum mechanics provides the theoretical basis of spectroscopy, which is the study of the interaction of electromagnetic radiation with matter. Spectrum is observed during transition in a state of a system and this transition from one energy level to the other is selective (selection rules),i.e., not all transitions are allowed which is a consequence of quantization (or discreteness) of energy as given by quantum mechanics ( time-dependent Schrödinger equation) .

21. Beside quantum mechanic has not only assisted chemistry but almost all other disciplines of science such as,physics,biology,medicine,computing and so on over the year evolving a better understanding of nature and has also precipitated a new block of super-smart real-time applications…..

22. Various structural properties are obtained using quantum mechanical interpretations of spectra, which help in structure elucidation. For the hydrogen atom, the quantum mechanical results so obtained successfully predict all aspect of the hydrogen atom spectrum. Quantum Mechanics & Chemical Bonding: A Chemical bond may be defined as the force that holds the atoms together in a molecules. However, the Schrodinger equation can not be solved exactly for a multi-electron system due to the presences of electron-electron repulsion terms in the Hamiltonian. The fundamental difficulty arises due to the fact that each electron repels every other electron so that the motion of each electron is dependent on the motion of all the other. However, solution of reasonable accuracy can be obtained using approximate methods. And henceforth, the chemical bonding in molecules is explained quantum mechanically via two popular theories Viz., Valence Bond Theory(VBT) and Molecular Orbital Theory(MOT).

23. Both these approaches assume a guess/approximate wave –function but the physical interpretation is different. Valence Bond Theory considers bond formation by overlapping of valence electrons in atomic orbital's and gives the concept of hybridization. Whereas, Molecular Orbital Theory describes bonding in terms of the combination and arrangement of atomic orbital's to form molecular orbital's that are associated with the molecule as a whole. This way the quantum mechanical treatment has been extended via more appropriate approximations to conjugated molecules (a popular approach to study the structure of conjugated molecules is via Hückel Molecular Orbital Theory), complexes and even polymers. The results/trends obtained from quantum mechanical approximations of various multi-electron atoms and molecules are in good agreement with the experimental results and it is for this reason that quantum mechanical wave-function interpretations have found acceptability. One needs to remember that the entire quantum mechanical interpretation lies in the solution of the Schrödinger equation which gives energy(particles character ) and associated with the microscopic matter. In fact the solution of the Schrödinger equation for wave function is the most difficult part as there is no unique method to solve it.

24. If the Schrödinger equation is solvable exactly , we get an exact wave-function for a system. But ,for multi-electron atoms or even for molecules, the exact solution of Schrodinger equation is not possible. In such cases , approximations are used where we build/guess an appropriate wave-function based on certain reasonable parameters and then try to solve the equation to obtain energy values. Once the wave-function is known , one can calculate any property of the system using appropriate quantum mechanical operators via methods such as eigenvalue equation or mean value theorem. The ultimate goal of quantum chemistry is to obtain wave-function si for a given system. Conclusively if one understands the basis of quantum mechanics, one can apply and solve any problem following the four-step quantum mechanical operation. Because of paucity of space and time, detailed mathematical formulation and more factual discussions about the subject are out of scope of this article. But one can say that quantum mechanics is essential for understanding every aspect of chemistry. Besides, quantum mechanics has not only assisted chemistry but almost all other disciplines of science such as, physics,biology,medicine,computing and so on over the years evolving a better

25. Understanding of nature and has also precipitated a new block of super-smart real time applications which includes ultra-precise clocks,un-crakable codes, super-powerful computers, improved microscopes, biological compasses ,GPS,lasers,telecommunications, smart phones,MRI scanners-the list goes on.

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