Hydrogenic Model:

The hydrogenic or hydrogen-like atom model is the most complex of the systems described so far. The reason it is called the hydrogen-like atom is because, while it contains only one electron like hydrogen, it may contain any number of nucleons in the atomic nucleus. The electron is present in a spherically symmetrical coulombic field, generated by the positively charged atomic nucleus. It is the common misconception that the nucleus remains stationary while the electron moves about it. In fact, both particles, if one can call them particles anymore, move about the atomic centroid, the center of mass for the atom. The motion of the electron needs to be studied as relative to this point and not the nucleus. Since it is present in a spherical field, the electron's coordinates relative to the centroid are, surprisingly enough, best shown in spherical coordinates. Also, the effective mass of the electron is changed from its true value, to a value known as the reduced mass, m , which is based on the center of mass coordinates of the electron.

eq.1

To begin solving for wave functions and eigenvalues, one starts with the Schrödinger Equation.

eq.2

The wave function must now be described in terms of the spherical coordinate variables: r, the distance of the electron from the centroid, f , the angle the radius vector makes with the x-axis when projected onto the xy plane, and q , the angle the radius vector makes with the z-axis.

Rewriting eq.2 explicitly in spherical coordinates:

eq.3

Notice the potential energy term on the left side which is dependent on 1/r. The Laplacian operator in spherical coordinates is difficult to derive, practically no QM texts show its explicit derivation. To begin solving this monstrous differential equation, the wave function y will need to be broken down into smaller functions. In the first of two reductions, y is broken down into its radial part and its angular part.

eq.4

Implementing eq.4:

eq.5

Notice in each term on the left, only one of the two functions gets differentiated. Dividing by RY and collecting the r²'s and the energy terms:

eq.6

Multiplying by -2 m r² / (h-bar)² and rearranging:

eq.7

In order for this to be true, both sides of this equation must equal each other for all values of r, f , and q. Therefore, each side can be set equal to a constant. This is the exact same procedure used when deriving the Schrödinger Equation. Concentrating on the angular factors first:

eq.8

Rearranging:

eq.9

The angular function must be broken down into two functions, one dependent on f , and the other on q :

eq.10

Implementing eq.10:

eq.11

Dividing by FQ , multiplying by sin² q , separating the factors dependent on f from those dependent on q, and implementing an arbitrary constant, m²:

eq.12

Concentrating on F first produces:

eq.13

This expression is similar to the one which appeared when deriving the Schrödinger Equation. The general solutions to F are:

eq.14

To find the value of N, we normalize:

eq.15

To determine the nature of m, we look at the boundary conditions of the function. Since it determines values as a function of the angle, f , we know that it is a periodic function:

eq.16

The constant m, is a quantum number, meaning it is influential in determining the state of the hydrogen atom system. When the function F is operated on by the L_z operator, the z-component of the atom's orbital angular momentum is found.

eq.17

The angular momentum is a vector, it is best for one to think of an arrow with its base attached at the atomic centroid. The quantum number m determines how far the angular momentum vector goes in the z-direction. If it were not for this, the orientation of the vector would be unknown.

Continuing on to find the other two functions, R and Q , and their corresponding eigenvalues.....

Returning to eq.12:

eq.18

One can neglect the right-hand side since it has been dealt with already.

eq.19

Multiplying by Q , dividing by sin² q , and setting m=0:

eq.20

As it was stated earlier in this site, many of the complicated differential equations encountered in quantum chemistry are solved by making them resemble equations previously solved. In the case of the angular function Q, eq.20 resembles the differential equation solved by Adrien-Marie Legendre. To transform it into a workable form requires a few substitutions:

eq.21

Utilizing our new substitutions:

eq.22

Expanding:

eq.23

To solve for the value of h, one can solve this equation by treating Q as a power series:

eq.24

Implementing this:

eq.25

Re-indexing the first term and rewriting the expression as one summation:

eq.26

In order for the above to be true, the quantity in brackets must be zero for all values of y:

eq.27

Rearranging, one arrives at a recursion relation, similar to the one obtained for the harmonic oscillator:

eq.28

Also, to simplify:

eq.29

Now for Q to be finite, this relation has to terminate. Let the value of n for which it does terminate be labeled l, (there's a reason for using l).

eq.30

The solutions to Q in the equation presented in eq. 23 are (Mortimer):

eq.31

The first bracket is a normalizing constant. The reason the letter l was used is because in classical mechanics, l represents angular momentum. The letter l is a quantum number like m, l is member to the set of whole numbers. The quantity l(l+1)*(h-bar)² equals the square of the total orbital angular momentum. It is derived from the L² operator:

eq.32

The square-root of l(l+1)*(h-bar)² produces the orbital angular momentum for the atom.

Now going way back to eq.7, one continues to find the remaining function R. Having the value of h :

eq.33

Rearranging, multiplying by R, and differentiating:

eq.34

Dividing by r² and multiplying out the quantity in brackets:

eq.35

To keep this from becoming too difficult to manipulate, substitutions are made:

eq.36

Implementing these:

eq.37

Since everything is in terms of r , one will want to convert the function R from being dependent on r to r . The transformations needed to change the previous expression:

eq.38

Implementing these:

eq.39

Dividing by 4*a ²:

eq.40

To ascertain the form of R, the asymptotic solution is taken, similar to the procedure used in the harmonic oscillator situation. The limit of the expression is taken as r approaches infinity:

eq.41

The general form of the asymptotic R is:

eq.42

One can state that the actual function R is the product of the asymptotic solution and another function which will be called F here.

eq.43

R will now be rewritten as R_InfF. Some expressions which will make this possible:

eq.44

Also:

eq.45

Implementing these:

eq.46

The exponential factors can be eliminated, since they never produce a value which would cause the overall expression to equal zero. Simplifing:

eq.47

Collecting terms of F:

eq.48

To ascertain the form of F, so one can figure out what R is, it must be broken down. One assumes F is:

eq.49

The exponent on r , c , is to be determined, L is an unknown function at this point. Some expressions which will be used:

eq.50

Also:

eq.51

Implementing these:eq.52

Expanding and then collecting terms of L:

eq.53

Dividing by r^c-2 and sorting the c 's in the last bracket by knowing A(A-1)+2A=A(A+1):

eq.54

Grouping the terms with r :

eq.55

To determine the value of c , the limit is taken as r approaches zero:

eq.56

Simplifing:

eq.57

This is beneficial. Now c can be replaced with l:

eq.58

This expression appears similar to the one solved by Edmund Laguerre, a French mathematician:

eq.59

Making eq. 58 conform to Laguerre's equation:

eq.60

Looking at the first brackets in each equation, one can draw similarities between the two:

eq.61

Also, looking at the second brackets:

eq.62

Laguerre found the solutions to L correspond to the following expression:

eq.63

One will notice by looking at this expression, that j and k must be natural numbers. Also, for L to be nonzero:

eq.64

The fact that l is less than or equal to k is important for realizing that beta in eq. 62 is equal to a positive integer. This integer can be represented as n, the quantum number used in the particle-in-a-box system. Symbolizing this:

eq.65

Using this, the full radial function R, can finally be constructed:

eq.66

The first bracket is a normalizing constant.

Rearranging eq. 65 so a is isolated, it can be compared with the value of a ² in eq. 36:

eq.67

Rearranging for energy produces the sought after energy expression which determines the energy states of the hydrogen-like atom system:

eq.68

This is dependent on Z, the number of charged nucleons in the atom, and n, the principal quantum number which is any integer greater than or equal to one.

Just as with the first two systems, the hydrogen-like atom possesses quantized energy states. An electron in this situation, will only make certain energy transitions, thereby emitting or absorbing only discrete amounts of energy. The overall wave function for the electron is dependent on three quantum numbers; n, l, and m. A generalized expression of the wave function can be written:

eq.69

Wave functions of single electrons are called orbitals, indirectly making reference to Niels Bohr's model of orbiting electrons. An orbital region is the region of space around the atomic nucleus which an electron is most likely to occupy. The principal quantum number n indicates the energy level an electron is in. It is any integer greater than or equal to one. The azimuthal quantum number l indicates the orbital angular momentum of the electron and therefore the shape of the orbital region. The magnetic quantum number m indicates the z-component of the orbital angular momentum and therefore indicates the orientation of the orbital region.

As discovered by P.A.M. Dirac, electrons possess a quantum analogue to intrinsic angular momentum, called spin. This fact is derived from relativistic quantum mechanics, not from the Schrödinger Equation. A fourth quantum number m_s indicates whether this intrinsic angular momentum is +h-bar/2 or -h-bar/2. Two electrons can occupy the same orbital or in other words have the same wave function, pending their intrinsic angular momentum projection vectors oppose each other.

The hydrogen-like atom model cannot be directly applied to polyelectronic systems. In such systems, the electrons exert repulsive forces against each other, greatly effecting the energy and orbitals of the system. Various approximation methods can be implemented to determine the states of such systems.