Basics to Quantum Chemistry: Operators, Functions and Properties of...

Greetings to everyone! Classical mechanics as formulated by Sir Isaac Newton is obeyed by macroscopic particles such as planets and rigid bodies. However, since the microscopic particles such as electrons, protons, atoms and molecules show wave-particle dual nature, and hence they do not obey Newtonian dynamics. They however obey Quantum Mechanics (or wave mechanics), a key feature of which is the quantization of energy and angular momentum. So, yes, I am going to start discussing quantum chemistry now. But since quantum chemistry is a vast and conceptual topic, it is better to start from some basic mathematics and mathematical tools used in quantum chemistry.

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Function

Basically a function is a rule that relates two or more variables. This mathematical constructs plays a pivotal role in representing the wavefunctions of electrons and the properties of molecular systems.

Suppose, z=x²+y², then z is a function of the variables x and y.
mathematically, z = f(x,y)

From previous studies on thermodynamics, we have learnt that volume, V is a function of pressure, P and temperature, T.

So, we can write, V = f(P,T)

This simply means that by varying values of pressure, P and/or temperature, T; value of V also varies. Thus, V is a function of T and P.

In quantum chemistry, the wavefunction is a central concept, representing the quantum state of a system. The wavefunction provides information about the distribution of electrons in a molecular system and serves as the foundation for calculating various properties such as energy, geometry, and reactivity. The wavefunction, denoted as Ψ, is often expressed as a mathematical function of the spatial coordinates of electrons. We shall discuss wavefunction in a separate blog, giving a rough idea about wave function should be sufficient for the discussions that we shall be doing for today.

Operator

Operators are pre-defined process or operation that changes one function to another function or constant entity. An operator is denoted by a 'hat' or a 'cap' (^).

Let us understand it by taking an example.

Just like we need minimum of two values for doing addition, subtraction, multiplication or division, likewise we need a function with an operator to draw physical significance. In the example above, we found the first derivative by doing operation on a function f(x).

Properties of operators

Operator is meaningless without a function.
Operators are directional in nature. i.e., they can be applied from left to right, but they will lose sense if the direction is reversed, say from right to left.
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The n^th power of operator means the operator will be operated n number of times.

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In quantum mechanics every operator is linear.

For an operator to be linear,

(a) Â {c f(x)} = c Â f(x), where c is a constant.

Let us see by taking an example.
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Therefore, d/dx is a linear operator. Similarly integration is also a linear operator. Can you tell me whether square root is a linear operator or not?

(b) Â {f(x)+g(x) = Â {f(x)} + Â {g(x)}

For every observable in classical mechanics, there is a operator in quantum mechanics. An observable is a variable that changes with the motion such as position, momentum, energy etc.

Let us see some of the important operators and their operation which will be frequently used in our studies.

Table 1) Classical observable and its quantum operator and operation

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Complete table can be found at, https://www.mathcha.io/editor/nKoEvu7QSeWtrL1l4ws6YN6Glcr90P8GHEOZkwQ
(All rights reserved to me)

Angular momentum operator in quantum chemistry

As an extension of the classical concept of angular momentum, the quantum mechanical counterpart introduces unique features that govern the dance of electrons around nuclei. The angular momentum operator, often denoted as L̂ is a vector operator in quantum mechanics. Its components, L̂_x, L̂_y and L̂_z, represent the projections of angular momentum onto the x, y, and z axes, respectively.

The three-dimensional angular momentum is a vector quantity and for a point particle it is classically represented as a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; p.

We can deduce the angular momentum operator for x, y and z directions based on the linear momentum operators we have learnt today.

Now, suppose r is a position vector having all the three dimensions.

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Problem) The angular momentum operator L̂_y is?

Solution

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What we learnt?

We were able to identify, define and understand what a function is! Functions are indispensable tools that allow scientists to model and understand the behavior of electrons and molecules. Whether describing the spatial distribution of electrons, calculating energy, or characterizing molecular orbitals, functions are the backbone of quantum chemical methods.
We learnt what a operator is in quantum mechanics and how it is derived from classical mechanics. We discussed the various properties of operator.
The angular momentum operator in quantum chemistry serves as a fundamental tool for understanding the behavior of electrons in atoms and molecules.