Why do different orbitals have different shapes?

P and S orbitals have different shapes defining what kind of bond they have like pi bond, sigma bond etc. So I was wondering what actually determines the shape and each region for different types of orbitals?




The simplest answer is, of course, that this is what solving the equations produces, which is not that helpful.


When the Schroedinger eqn. for the H atom is solved instead of using $x,y,z$ coordinates we use $r,\theta,\phi$ as mathematically the solution is easier this way, where $r$ is radial distance and $\theta, \phi$ angles. As well as radial solutions, depending on distance from nucleus $r$ (and $n,\ell$ quantum numbers), there are also mathematically well known angular functions called spherical harmonics which can be complex numbers and have various shapes depending on the quantum numbers. To be able to draw these, and to make chemical sense, we have to combine them together so that they are no longer complex numbers, and this is what is done to form the p orbitals etc. Mathematically this 'trick' is ok as the sum of solutions can be shown also to be a solution of the Schroedinger eqn.


So its not simple, but once we have the orbital shapes we can then begin to join them together to form bonds or even combine them again to form hybrids such as tetrahedral $sp^3$.


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> 
>  A solution to the Schrödinger equation maybe written as a product of radial and spherical functions. The size of different orbitals varies according to the radial function and the shapes of different orbitals are a result of different spherical functions.
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### The Schrödinger equation


You might be familiar with the following (magic) equation:


$$
\hat{H}\mathrm{\Psi} = E\mathrm{\Psi} 
$$


where $\hat{H}$ is the *Hamiltonian operator*, and $\mathrm{\Psi}$ is the *eigenfunction* corresponding to its *eigenvalue*, which is, energy $E$ of the system. $\mathrm{\Psi}$s, often called wavefunctions, are obtained by solving the Schrödinger equation.


### The Born Interpretation


The different orbitals are actually *isosurfaces* of the square-magnitudes $|\mathrm{\Psi}|^2$ of these $\mathrm{\Psi}$s:


$$
|\mathrm{\Psi}|^2 = \mathrm{\Psi}^\*\mathrm{\Psi}
$$


which, for an electron, represent the probability of finding the electron on that surface. Mostly, the maximum-probability isosurface is shown.


### Separation of Variables


$\mathrm{\Psi}$ is a function of space-coordinates and time. We may represent space-coordinates in radial form $(r\text{, }\theta\text{, and }\phi)$ and apply *separation of variables*:


$$
\mathrm{\Psi}(r, \theta, \phi, t) = R(r)Y(\theta, \phi)f(t) \tag{1}
$$


where, the radial component, $R(r)$, of $\mathrm{\Psi}$, roughly speaking, represents *size* of the orbital, the angular component, $Y(\theta, \phi)$, (again) roughly speaking, represents *shape* of the orbital, and we also have a time component $f(t)$ to represent how this wavefunction varies with time. From Equations (1) and the Schrödinger equation:


$$
\hat{H}(R(r)Y(\theta, \phi)f(t)) = E(R(r)Y(\theta, \phi)f(t)) \tag{2} 
$$


### Spherical Harmonics


We need to focus on the shape of orbitals; therefore, our discussion will be on $Y(\theta, \phi)$. When we solve Equation (2) for $Y$, we obtain spherical harmonics which are dependent on two variables called $l$ and $m$. Yes! these correspond to the two principal quantum numbers of the four $n$, $l$, $m$, and $s$, where $l\in [0,n-1]$ and $m\in [-l,l]$:


$$
Y(\theta, \phi) = Y\_l^m(\theta, \phi)
$$


The solutions, if visualized, look (something) like this:


[![comparison between spherical harmonics and shape of orbitals](https://i.stack.imgur.com/8505u.jpg)](https://i.stack.imgur.com/8505u.jpg)


Notice how the shape of orbitals emerges naturally from the sign and values of spherical harmonics.


### Reference


Levine, I. N. (2014). *Quantum Chemistry*, *7*$^\textit{th}$ *edition*. Pearson Education Inc.


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