Construction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems

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1 Construction of the Electronic Raial Wave Functions an Probability Distributions of Hyrogen-like Systems Thomas S. Kuntzleman, Department of Chemistry Spring Arbor University, Spring Arbor MI 498 Copyright Tom Kuntzleman 6. All rights reserve. You are welcome to use this ocument in your own classes but commercial use is not allowe without the permission of the author. Goals:. This ocument attempts to illustrate a metho by which one may generate an present graphs of the raial electronic wave function an probability istribution for hyrogen-like systems "from scratch". No attempt is mae to erive this wave function.. This ocument is intene to provie the user with an unerstaning of the origin, form an characteristics of the Laguerre an associate Laguerre polynomials, which are use in the construction of the raial electronic wave function of hyrogen-like systesm. This unerstaning shoul enable the user to communicate why the quantum number l can only have values from to n -. Objectives: After completing this exercise, stuents shoul be able to:. Separately evaluate an graphically isplay the ifferent functions of r (where r is the istance of the electron from the nucleus) that are use in constructing the raial electronic wave function for hyrogen-like atoms an ions, given the quantum numbers n an l.. Construct an plot the raial electronic wave function for hyrogen-like systems an its associate probability ensity, given any values of the quantum numbers n an l.. Determine the most probable istance of the electron from the nucleus for at least 5 atomic orbitals. 4. Investigate the relationship between the number of noes in an electronic raial wave function of H-atom-like systems an the quantum numbers n an l. 5. Describe why the quantum number l must be smaller in magnitue than the quantum number n, accoring to the associate Laguerre polynomial use in constructing the electronic raial wave function of H-atom-like systems. Introuction: The raial electronic wave function of the H-atom is compose of four parts:. A normalization constant, N.. A power function of r (where r is the istance of the electron from the nucleus), ρ, page

2 where ρ = r na, a is the Bohr raius, an l is the angular momentum quantum number.. An exponentially ecaying function of r, e -ρ/. L ( ρ) 4. An associate Laguerre polynomial, l where n is the principal quantum n l number. Altogether, the wave function has the form: R N e L n l ρ / l ( ρ) = ρ l ( ρ) This ocument will focus on the generation of these iniviual functions an a stuy of their properties. In aition, these separate functions will be combine so that graphs of the electronic raial wave function of the H-atom an similar ions may be isplaye. Laguerre Polynomials We begin by stuying the Laguerre polynomials an show how they are obtaine an examine their properties. The Laguerre polynomial is a built-in Mathca function. For example, to call the Laguerre polynomial that is n orer in x, you woul type Lag(,x). You can observe the characteristics of the first five Laguerre polynomials below in Graph. Note carefully the shape of each curve an ientify the function type of each line, e.g. linear. Be sure to ientify the orer of the polynomial with the curve function type. Graph :The First 5 Laguerre Polynomials Lag(, x) Lag(, x) Lag, x Lag, x Lag( 4, x) 4 5 x Question : Graph the 7th orer Laguerre polynomial. To o so, click somewhere in the space below so that you see a re plus sign. Then click on the X-Y graph button on the menu bar above. (Alternatively, you may simultaneously type shift). Next, type Lag(7,x) in the box irectly to the left of the y-axis (the box in the mile). Create extra space below by pressing the enter key several times. page

3 Alternatively, the i th Laguerre polynomial may be generate from the following expression: e x i x i e x i! i x expression This is the most general metho for obtaining Laguerre polynomials which we will use in the remainer of this ocument. Note the ivision by i!. This is use to insure that the simplest form of any particular Laguerre polynomial is generate. Division by i! is also necessary to be certain that the Laguerre polynomial generate by expression is the same as Mathca's built-in Laguerre function (but see footnote at the en of this ocument). Question : Evaluate expression for i = 7 by replacing each "i" with "7" in expression () above. In the expression for the erivative, replace the i in the lower portion of the erivative expression with a 7 first -- the i in the upper portion will then automatically be change as well. Once you have complete this, highlight the above expression, click on "Symbolics" an then "Simplify". Mathca shoul have returne the following polynomial, which is equal to the built-in Mathca function, Lag(7,x): 7x x 5 6 x 5 4 x4 7 4 x5 7 7 x6 54 x7 We're going to check to see if we have generate the 7th orer Laguerre polynomial using expression. Define F(x) an G(x) as the 7th Laguerre polynomial as follows: F( x) := Lag( 7, x) G( x) := 7x x 5 6 x 5 4 x4 7 4 x5 7 7 x6 54 x7 Here is a plot of both F(x) an G(x): page

4 Graph : A comparison of polynomials Fx Gx If the curves match up, you've got the right function! 4 5 x Question : Generate the 8th Laguerre polynomial (use Mathca's built-in Laguerre function. Define this function as F(x), an make a plot of F(x). Next, generate the 9th Laguerre polynomial function using expression, an efine this function as G(x). Overlay G(x) onto the plot of F(x). To overlay this secon plot, click on the "F(x)" that is paste to the left of the y-axis. Hit the space bar until the entire "F(x)" is unerline. While the F(x) is unerline, hit the comma button on your keyboar. A new space shoul present itself on the y-axis into which you may paste "G(x)". Notice the shape of each function. Start to think in terms of well behave solutions to the raial ifferential equation. Associate Laguerre Polynomials. During the erivation of the complete solution to the raial function of the hyrogen atom, one iscovers that the polynomial require is a solution to the Associate Laguerre ifferential equation. Details of the erivation can be foun in Eyring, Walter an Kimball or Pauling an Wilson liste in the references at the en of this ocument. It turns out that the raial function epens upon the quantum numbers n an l through the associate Laguerre polynomials, which may be efine as: j L i (x) = j L ( x) j i x th, where L ( x) is the i Laguerre polynomial. i where i = n l an j =l. In other wors, the j th associate Laguerre polynomial of orer i is simply the polynomial you get after taking the j th erivative of the i th orer Laguerre polynomial. i an j are efine in terms of the hyrogen atom quantum numbers n an l (i = n l an j =l ). The associate Laguerre polynomial for small values of i an j may be foun by taking erivatives of the built-in Mathca Laguerre polynomial function (type Lag(n,x)). You may notice, however, that it is ifficult for Mathca to compute the associate Laguerre polynomial for higher values of i an j. Define i an j i := j := page 4

5 Graph : Associate Laguerre Polynomial j Lag( i, x) j x 5 5 x Later we will see that not all values of i an j will arise when i an j are efine in terms of the n an l hyrogen atom quantum numbers. Question 4: Fin the 5 th associate Laguerre polynomial of orer 7 by replacing the appropriate values for i an j, highlighte in yellow above. This is equivalent to asking for the associate Laguerre polynomial for n = 5 an l =. Was Mathca able to evaluate the result? (If so, you will see a plot in Graph above). Now Fin the 7 th associate Laguerre polynomial of orer 8 by replacing the appropriate values for i an j. Was Mathca able to evaluate this result? Clearly, we nee a ifferent metho to fin associate Laguerre polynomials for large values of i an j. We'll fin the associate Laguerre polynomial for i = 8 an j = 7, as follows. First, fin the 8 th Laguerre polynomial by substituting i = 8 into the expression below, then clicking "Symbolics" an "Simplify". Recall that j = l an i = n l an thus we are ealing with a 5f orbital. Laguerre Polynomial Generating Function e x i x i e x i! i x expression () Next fin the 7 th erivative with respect to x of the 8 th Laguerre polynomial you foun above. Click somewhere in the space below. From the menu bar, choose the n /x n button from the menu bar. (You may have to choose View --> Toolbars --> Calculus first). You can alternatively type CtrlShift/ to access the erivative operator. Copy an paste the 8 th Laguerre polynomial (evaluate above) into the box you see on the far right han sie of the n /x n notation. Choose the 7 th erivative with respect to x. Then highlight the expression you have generate, click "Symbolics" an "Simplify": Di you get -8 x? If not, repeat the steps above until you o. page 5

6 Now efine F(x) as L 7 8 (x) (paste the appropriate function in the right han sie of the space below: Fx := Plot the resulting associate Laguerre polynomial. You shoul notice that any associate Laguerre polynomial is (i-j) th orer in x ( in this example i = 8 an j = 7): Fx 5 5 x Question 5: Plot the associate Laguerre polynomial with i = an j =. Which type of orbital oes i = an j = represent? What woul happen if you trie to plot the associate Laguerre polynomial with j > i? Why? As state previously, the solution to the raial portion of the H-atom inclues an associate Laguerre polynomial. The associate Laguerre polynomial use in the H-atom wave function epens upon two integers (quantum numbers for the H-atom). The associate Laguerre polynomial is efine in terms of j = l an i = n l as follows: L l nl (r). In other wors, for a 4p orbital (n = 4 an l = ), we' nee to fin the r (*) erivative of the 5 th (4 ) Laguerre polynomial. We will now evaluate the associate Laguerre polynomial for a 4p orbital. First, efine n = 4 an l = : n := 4 l := This means i an j for the appropriate associate Laguerre polynomial in the 4p wave function of the H-atom must take on the following values: i := n l j := l i = 5 j = Now we can fin the appropriate associate Laguerre polynomial for a 4p orbital. We efine this associate Laguerre polynomial as La( ρ), a function of the istance r of the electron from the nucleus, below: La( ρ) j r := Lag( i, ρ) Recall that ρ =. j ρ na Here is a plot of our associate Laguerre polynomial for a 4p orbital: page 6

7 Graph 4: Associate Laguerre Polynomial La( ρ) 5 5 ρ You may notice some "noise" in Graph 4 for some values of n an l. Remember, if Mathca is having trouble evaluating l L ( ρ) n l j using the built-in function then no plot at all will be isplaye in Graph 4 jlag(i,ρ), r above. Vary the values of n an l to etermine the range of applicability of the general approach shown above. You shoul iscover that l cannot be greater than meaning that only up to orbitals can be stuie for any n value. If you cannot use the built-in function, you will nee to manually evaluate an paste (in the right han sie of the expression in yellow, below) your chosen associate Laguerre polynomial as outline above -- by generating the appropriate Laguerre polynomial using expression an then taking its proper erivative. The Lag function only returns a value of the Laguerre polynomial at point ρ it is not the most general approach to use to examine these functions systematically. This may be the cause of the ifficulty with obtaining certain plots. Thus we must etermine a set of Laguerre an associate Laguerre polynomials for use in the remainer of the ocument. The first few are given in the table below. Your job is to complete the table for missing polynomials. A sample calculation for n= an l= is given in the following collapse region. Sample calculation of an array entry Each Laguerre poynomial must be generate inepenently to complete the table. Laguerre Polynomial Generation Li( i, x) e x i := x i e x Generating function i! i x page 7

8 Specific Function Li(, x) e x e x xe x Li(, x) ex e x 4xe x x e x Extracte Result Simplifie Result e x e x xe x x ex e x 4xe x x e x x x For the associate Laguerre polynomials we introuce the use of n an l since we will nee them later as the hyrogen atom quantum numbers. Note the use of the operator for generating the associate Laguerre polynomial. This is followe by extract an then simplify to get the final result. The approach is very general. n := l := i := n l j := l i = j = operate j e x i x i e x j x i! i x ex e x 4xe x x e x ex ( 6) e x 6xe x x e x extract simplify ex e x 4xe x x e x x ex 6e x 6xe x x e x page 8

9 n l i 5 j x 5x 5x Laguerre x x x 5 x x 6 x 5 4 x4 x5 AssocLaguerre x 5x x Define the associate Laguerre polynomial as La( ρ) for your chosen values of n an l, by pasting the correct function into the right han sie of the efinition below. For n = an l = La ρ r := ρ Recall that ρ =. na Graph 5A: Associate Laguerre Polynomial Plot Graph 5B: Laguerre Polynomial Plot La( ρ) ρ ρ 5.5 ρ 5 ρ Prepare several plots of associate Laguerre an Laguerre polynomials to see the relationship between the two. page 9

10 Normalization Factor Now that we have efine the associate Laguerre polynomial for several values of n an l, we can begin to buil other portions of the H-atom raial wave function. First, the normalization constant for the raial portion of the H-atom wave function epens upon n an l. This constant is efine as N below, an then evaluate for the values of n an l : n := l := N( n, l) := ( n l )! n( ( n l)! ) Nn (, l) =.4 or N(, ) =.4 You can vary n an l to fin any normalization factor you wish. Note that two ways you can o this are shown here... Before putting the entire wave function together for the raial portion of the H-atom, we will look at two more functions of r that are inclue. The first of these is the function efine as φ(r,n,l) below. This function is l th orer in r. In the equation below, a is efine as the Bohr raius. In aition, To exten this an other equations from escribing only the H-atom to hyrogen like ions (such as He or Li ), we also inclue the nuclear charge, Z: φ ( r, n, l) := a := 5.9 Z := Z na l r l φ(r) an is isplaye below, base on the values of n an l you efine most recently in this worksheet. You may nee to ajust the parameter "max" in orer to fully view the function. The function is multiplie by a / to keep it unitless in the graphical isplay. Because max is efine as below, the wave function will be isplaye from to a : max := page

11 Graph 6: r^l factor a φ( rn,, l) Question 6: What changes o you observe in φ(r) when the values of n an l are change? (Just change the n or l in the φ(r,n,l) in the y axis to what ever value you wish to examine). Specifically, how oes φ(r) change when l =? How about when l >? Why oesn't this function significantly change with changes in n? r r / m page

12 The secon function of r is an exponentially ecaying function of the istance of the electron from the nucleus. It is efine as EXP(r) an isplaye below: EXP( r) := exp Z r na Graph 7: Exponential factor EXP( r) r r / m Question 7: Vary EXP(r) by changing the values of n. As you increase n, oes EXP(r) approach zero at larger or smaller values of n? Does this harmonize with what you know about the size of orbitals as n increases? Observing the functions we get when we multiply φ(r,n,l) an EXP(r) together gives some insight into the nature of raial H-atom-like orbitals with ifferent values of n an l. These functions are multiplie together an efine an isplaye as Ρ(r,n,l), below: Ρ ( r, n, l) := φ ( r, n, l) EXP( r) Graph 8: phi(r)exp(r).5 a Ρ( rn,, l) r r / m page

13 Question 8: a. What changes o you observe in Ρ(r) when the values of n an l are change? Specifically, how oes Ρ(r) change when l =? How about when l >? What oes this tell you about the raial wave function of s orbitals as compare to other orbitals? b. Is it φ(r) or EXP(r) which ensures that the probability of fining the electron at large istances from the nucleus is vanishingly small? Is it φ(r) or EXP(r) which ensures that the wavefunction vanishes at the nucleus when l >? c. Each place an orbital curve crosses the r-axis is calle a noe (notice that the wave function oes not cross the r-axis at r = ). Do you observe any noes in Ρ(r) at any values of n an l at all? Base on this observation, o the noes in certain H-atom wave functions originate from φ(r), EXP(r), or La(r)? You may nee to revisit the associate Laguerre polynomials before answering this question. Finally, the complete wave function for the raial portion of the H-atom can be efine (as R(r), below). The wave function inclues the normalization constant (N), a function of r l (φ(r)), an exponentially ecaying function (EXP(r)), an the associate Laguerre polynomial (La(Zr/na )). It is also necessary to multiply R(r) by i! = (nl)! (see footnote at the en of this ocument). You shoul notice that La(Zr/na ) is a function of r just like φ(r) an EXP(r). Also notice that the Laguerre polynomial is liste as a function of Zr/na. As a reminer, r is the istance of the electron from the nucleus, a is the Bohr raius, Z is the nuclear charge an n an l are quantum numbers: The raial wave function for hyrogen-like systems: n = l = The proper associate Laguarre polynomial must be obtaine from the ata array table an paste into the raial function. LA _ := NOTE: Define HERE the associate Laguerre polynomial (LA n_l ) that you want to use for the orbital an probability plots (see below). simply cut an paste from the maxtrix array of the above. See footnote (at the en of this ocument) for a iscussion of why it is necessary to multiply by (nl)!. R( r, n, l) := Nn (, l) Ρ ( r, n, l) LA _ ( ( n l)! ) R(r) is plotte below, using the values of n, l, Z, an the associate Laguerre polynomial you efine above. Because R(r) has units of a /, we multiply the wave function by a / to make it a unitless quantity. You can ajust the maximum istance over which you observe the raial wave function by ajusting the parameter "max": page

14 max :=. Graph 9: Raial Orbital Plot. a Rrn (,, l) The square of the raial wave function gives you a quantitative escription of the probability of fining the electron at a istance r from the nucleus. R(r) is multiplie by r in orer to present this probability istribution as a function of a spherical shell aroun the nucleus. Furthermore, to make r R(r) unitless, we nee to multiply by a (can you verify this?): r max :=. Graph : Probability Distribution a r Rrn (,, l) r page 4

15 Concluing Questions: 9. Plot R(r) an r R(r) for your choice of 5 ifferent orbitals. You shoul also view the results from other groups that plot ifferent orbitals than the 5 your group chooses. These plots can be mae by changing the values for n an l, as well as the associate Laguerre polynomial (LA n_l ) that are highlighte in yellow or pink further up on this worksheet. However, you will nee to manually fin the erivative of the Laguerre polynomial for the 4f an 5f orbitals, an efine this manually etermine polynomial in the appropriate place further up on this worksheet.. Use Graph 9 to estimate the most probable istance of the electron from the nucleus (r mp ) for 5 orbitals of your choice in units of a. For example, r mp for a 4p electron is about a. How oes r mp change for each of these orbitals if the nuclear charge is increase to Z =? Does this make sense? Why or why not?. Each place the R(r) (Graph 8) crosses the r-axis is calle a noe (notice that the wave function oes not cross the r-axis at r = ). Can you etermine a relationship between, n, l, an the number of noes in an orbital?.. You probably have learne that the quantum number l cannot be greater than or equal to n (l =,,,... n - ). The associate Laguerre polynomial use in constructing the raial function of the H-atom electron is etermine by L l nl (r). Accoring to this associate Laguerre polynomial, why is it that n - > = l? EXPERT QUESTION:. Display graphs of the p orbital, but without the following portions of the H-atom wave function inclue: a. The normalization constant (multiply by - to get the phase right) b. φ(r) c. La(r) (multiply by - to get the phase right). EXP(r) Be sure to isplay the r-axis from a minimum of in each case. Briefly escribe the effect of each omission on the characteristics of the p orbital. State clearly why each component is require in the raial function. page 5

16 Acknowlegment: I wish to thank Theresa Julia Zeilinski for valuable insight, ieas an guiance in the evelopment of this ocument. References : L Pauling an E. B. Willson, Introuction to Quantum Mechanics, McGraw Hill, NY, 95. H. Eyring, J. Walter, an G. E. Kimball, Quatum Chemistry, John Wiley an Sons, Inc. NY, 944. D.A. Simon an J.D. McQuarrie, Physical Chemistry: A Molecular Approach, University Science Books, Sausalito, CA, 997. Footnote: In this worksheet, a comparison is mae between the Laguerre polynomial generate by Mathca (Lag(n,x) an the Laguerre polynomial generate by the expression (See graph ): e x i x i e x Expression i! i x The above generating expression was chosen to match the Lag(n,x) function in Mathca so that the equivalence between these methos of generating Laguerre polynomials may be observe. However, the conventional generating function for a Laguerre polynomial is ifferent from the Lag(n,x) function in Mathca: e x i x i e x Expression i x Notice that Expression is simply Expression multiplie by i!. Because Expression is use in generating the Laguerre polynomials in this worksheet, an because i = nl, it is therefore necessary to multiply the overall raial wavefunction by ( nl)!. page 6

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