The equations and laws of physics are not so simple to understand, their description of the real world can seem pretty complex in nature. To provide a perfect analytical solution to every problem is not possible, unless we can set up laws that can take every variable involved in the problem into consideration. However, basic laws can still be useful if one can inculcate the real world variables, in terms of approximations. Calculating the orbitals of a multielectron atom is analytically not possible, using the Schrodinger equation. However, with the use of the concept of effective nuclear charge, a rough solution can be obtained.
As you may know, like charges repel each other, while unlike charges attract. A positive charge will attract negative charges, while repelling other positive charges with the electromagnetic (Coulomb) force. Similarly, negative charges will attract other positive charges, while repelling other negative charges. What binds the electron cloud to the atomic nucleus is the attractive electromagnetic force between electrons and protons inside the nucleus.
The simplest and most completely understood atom is that of Hydrogen. It's inherent simplicity, implied by the fact that it consists of one electron revolving around a single proton, makes it easy to calculate its atomic orbitals. The Schrodinger equation for the Hydrogen atom is exactly solvable. However, for atoms with more number of electrons, calculating the exact atomic orbitals is not analytically possible. Hence, the concept of effective nuclear charge is used to get a rough solution.
It is defined as the effective positive charge felt by an electron in the outer shells of an atom, due to the screening effect of electrons in the inner shells. The electrons in an atom are distributed in concentric shells around the nuclear center. The attraction felt by outermost electrons, because of the positive nuclear charge, is lessened by the repulsive force exerted by inner shell electrons. It is the result of subtraction of average number of inner shell electrons, from the total atomic number. The calculation formula is the following.
Effective Nuclear Charge (Zeff) = Z - N
where Z is the atomic number and N is the number of inner shell electrons. So the outer (valence) electrons of an atom or an ion, see the inner nuclear charge to be lesser than what it actually is. The consideration of effective nuclear charge makes approximate analytical solutions for multielectron atoms possible.
To calculate, there are only two things that you need to know. One is the atomic number and second is the electronic configuration. Actually, if you know the atomic number, then the ground state electronic configuration can be guessed.
Let's consider the example of Carbon, whose atomic number is 6. Its electronic configuration is 1s2 2 s2 2 p2. Since Z = 6 and N = 2, a valence electron in the 'p' shell will have an effective nuclear charge of '6 - 2', which is equal to 4.
All that it takes to calculate the effective charge of any atomic nucleus is an understanding of the electronic configuration of the atom in the ground state. This makes it easier to make calculations.