Quantum Chemistry for the Beginners

Lesson II :: Quantum Chemistry of Atoms

Lesson I :: Basic Concepts of Quantum Chemistry

2.1. The simplest case of the hydrogen atom and the hydrogen-like atomic systems: 

The hydrogen atom, with one proton acting as the atomic nucleus and only one electron, is obviously akin to some other particular atomic ions (He⁺ ion, Li²⁺ ion, Be³⁺ ion etc.), as each of the above contains only one nucleus (that within it containing Z number of protons, where Z -- e.g., 1, 2, 3, 4 -- is the atomic number) and only one electron. So the H-atom and the aforesaid one-electronic atomic ions are together called 'hydrogen-like atomic systems'. Having only one nuclei and only one electron, such simple systems provide exact, direct solutions of the Schrodinger equation. To tell you the complete truth, for any such systems the complete Schrodinger equation is actually separable into two simpler Schrodinger equations -- one for the translational motion (i.e., for movement from one location to another) of the atom/ion/molecule as a whole, and the other one for the motion of the electron and the nucleus relative to each other (this, however, can be simply visualised as just the motion of the electron relative to the nucleus -- the electron being much lighter than the nucleus). The latter Schrodinger equation, for the electronic motion, can be simply stated as ĤY = iħ(∂Y/∂t), with the corresponding time-independent Schrodinger equation being stated as Ĥy = Ey (E being the electronic energy of the atom), where the Hamiltonian operator Ĥ for a hydrogen-like system is simply {–ħ²/(2m_e)}.{(∂²/∂x_e²) + (∂²/∂y_e²) + (∂²/∂z_e²)} – Ze²/(4pe_or_e). It should be noted that here y = y(q) i.e., y(x_e,y_e,z_e) or y(r_e,q_e,f_e) in the last equation is the electronic wavefunction, and that an electronic wavefunction for a one-electron atomic system is also called an orbital (say, the 1s orbital). [However, for a multi-electron atom, it so happens that the electronic wavefunction is a sort of a product of several (occupied) electronic orbitals -- that idea is detailed later.]
Note: Have you noted the term 'atomic ion'? An atomic ion (whether cation or anion, e.g., Na⁺ or Cl^–) can be formed from one atom by loss or addition of only electron(s), and is thus distinct from molecular ions (e.g., H₂⁺, N₂^–) and multi-atom radical ions (e.g., NH₄⁺, SO₄^{2 –}).

For sake of truth, it may now be told to you (should it really be?) that the above use of m_e (mass of electron) in the factor {–ħ²/(2m_e)} multiplying the Laplacian is only an approximation, which quantity should, exactly speaking, be replaced with the 'reduced mass' m equalling m_e.m_nuc/(m_e+m_nuc) -- but this simplification introduces rather too minor an error (as obviously m ≈ m_e -- can you recognise how?). More importantly, note the appearance of Z in the potential-energy expression –Ze²/(4pe_or_e), which was absent (there Z was unity!) in the earlier expression in Lesson I given about only the H-atom. Here, however, we give a general expression true for H-atom, He⁺ ion, Li²⁺ ion etc. i.e., for all one-electron one-nucleus systems, and so here Z does appear [here (–e) is the charge of the electron, (+Ze) is the charge of the nucleus, their distance is r_e, and so the potential-energy expression, as per Coulomb's law, is –e.Ze/(4pe_or_e)]. Further, we choose the electron to be at the point (x,y,z) -- obviously equivalent to (r,q,f) -- while the nucleus is chosen to be at the origin O of the coordinate system. This means that (∂²/∂x_e²) + (∂²/∂y_e²) + (∂²/∂z_e²) can be simplified as being (∂²/∂x²) + (∂²/∂y²) + (∂²/∂z²) while –Ze²/(4pe_or_e) can be simplified as –Ze²/(4pe_or). Now expressing the Laplacian i.e., (∂²/∂x² + ∂²/∂y² + ∂²/∂z²) as ∂²/∂r² + (2/r)∂/∂r + (1/r²)∂²/∂q² + (1/r²)cotq.∂/∂q + (1/r²)cosec²q.∂²/∂f², we get the electronic Schrodinger equation fully spelt out in terms of (r,q,f) coordinates only, and can now easily proceed to solve it. 

2.2. How did we arrive at the familiar hydrogen-atom electronic wavefunctions?  

The aforesaid (time-independent) electronic Schrodinger equation (expressed in (r,q,f) coordinates) can be further separated into two daughter differential-equations -- a radial equation (involving only r coordinate) and an angular equation (involving only q and f coordinates -- I don't want to tell what exactly are these two equations). The radial equation when solved (via a complicated procedure which you better be ignorant of) gives a function of r as its solution -- this function is called the radial wavefunction, denoted as R_nl(r). The angular equation when solved (via a comparably complicated procedure) gives a function of q and f as its solution -- this function is called the angular wavefunction or the 'spherical harmonics', denoted as Y_lm(q,f). The complete electronic wavefunction for the hydrogen-like atomic system, Y_nlm(r,q,f) is the product of the radial and the angular wavefunctions, i.e., Y_nlm(r,q,f) = R_nl(r).Y_lm(q,f) [the wavefunction Y_nlm is the eigenfunction of the Hamiltonian operator in the aforesaid electronic Schrodinger (time-independent) equation with the energy eigenvalue E = –(13.6Z²/n²) eV]. You are probably guessing by now (yes, this is indeed so) that the subscripts n, l and m means, respectively, the principal, the azimuthal and the magnetic quantum numbers of the atomic orbital (occupied by the lone electron in the hydrogen-like atomic system). From the symbolism used, it is clear to you that the radial wavefunction depends only on the principal and the azimuthal quantum numbers, whereas the angular wavefunction depends only on the azimuthal and the magnetic quantum numbers. Also, there is nothing mysterious about these three quantum numbers appearing in atomic electronic wavefunctions with definite rules (e.g., l < n) binding them -- these quantum numbers, along with the well-known limitations binding their values, arise simply while solving the radial and the angular equations (in a process similar to the appearance of integral quantum-number - n - values for the aforesaid particle in a one-dimensional box system). 

The radial and the angular wavefunction must remain separately normalised, so that the complete wavefunction Y_nlm(r,q,f) remain normalised. This means that the normalisation requirement ₀∫^+∞₀∫^2p₀∫^p |Y_nlm(r,q,f)|² r² sinq dq df dr = 1   i.e., {₀∫^+∞|R_nl(r)|².r² dr}.{ ₀∫^2p₀∫^p |Y_lm(q,f)|² sinq dq df} = 1, should imply two specific sub-requirements (i) ₀∫^+∞|R_nl(r)|². r² dr = 1 and (ii) ₀∫^2p₀∫^p |Y_lm(q,f)|² sinq dq df = 1. It is these two requirements that decide what'll be the exact forms of these two wavefunctions. As an example, the 2p_o hydrogen-like wavefunction (i.e., Y₂₁₀) is {Z⁵/(32pa_o⁵)}^1/2 r exp{–Zr/(2a_o)}.cosq, and for this complete wavefunction the radial and the angular wavefunctions (i.e., R₂₁ and Y₁₀) are, respectively, {Z⁵/(24a_o⁵)}^1/2 r exp{–Zr/(2a_o)} and {3/(4p)}^1/2 cosq. [In other words, they can't be something like {Z⁵/(32pa_o⁵)}^1/2 r exp{–Zr/(2a_o)} and cosq respectively -- check for yourself whether both the normalisation requirements are being separately satisfied by the former (i.e., the valid) pair of radial and the angular wavefunctions.]
Problem: Find the angular and the radial wavefunction corresponding to the hydrogen-like 1s-state wavefunction Y₁₀₀. Ans: We knew (Section 1.5) that Y₁₀₀ = ({Z³/(pa_o³)}^1/2 exp(–Zr/a_o), and now we know that Y₁₀₀(r,q,f) = R₁₀(r).Y₀₀(q,f). However, this function Y₁₀₀ has no dependence on q and f, and so its (q,f) dependent part -- i.e., Y₀₀(q,f) -- must be nothing but a constant. Let it be N_Y, a real positive constant -- just like a normalisation constant. This means that the aforesaid normalisation requirement for Y_lm(q,f) function simply implies ₀∫^2p₀∫^p N_Y² sinq dq df = 1, i.e., N_Y² = 1/₀∫^2p₀∫^p sinq dq df. From earlier experience (Section 1.5), we know the integral at the denominator to be 2 x 2p = 4p, implying that N_Y = (1/4p)^1/2 i.e., Y₀₀(q,f) = (1/4p)^1/2. Now, utilising the relation Y₁₀₀(r,q,f) = R₁₀(r).Y₀₀(q,f), we may immediately get R₁₀(r) = Y₁₀₀(r,q,f)/ Y₀₀(q,f) = 2(Z³/a_o³)^1/2 exp(–Zr/a_o) -- isn't it?

2.3. The forms of the common radial and angular wavefunctions:  

Let us now put forward the mathematical expressions (check the already known ones) of some of the radial and angular wavefunctions (i.e., R_nl and Y_lm) as well as the complete wavefunctions Y_nlm(r,q,f) associated with all the hydrogen-like atomic orbitals corresponding to the 1s, 2s, 2p, 3s, 3p and 3d subshells. We'll then take note of some of their interesting characteristics.

Table 2.1: The Radial Wavefunctions R_nl(r)

Table 2.2: The Angular Wavefunctions Y_lm(q,f)

*Note:  exp(if) = cosf + i sinf,   exp(–if) = cosf – i sinf    where   i = (–1)^1/2.  Also note that
Y_lm(q,f) is always of the form Y_lm(q,f) = g(q).exp(±imf), where g(q) is a function of q only.

Table 2.3: The Complete Wavefunctions Y_nlm(r,q,f) = R_nl(r).Y_lm(q,f)

For wavefunctions or orbitals with an non-zero value of the magnetic quantum number m, such as 2p₊₁ and 2p_–1 displayed above, we find that the wavefunction is not real, i.e., complex (e.g., 2p₊₁ above contains a complex factor cosf + i.sinf). Also, for these orbitals the directional characteristics are not much specific, e.g., the 2p₊₁ function is not cylindrically symmetric around the x axis or the y-axis, unlike the 2p_o function (also known as 2p_z) which is cylindrically symmetric around the z-axis (you probably intuitively follow what the last clause means). So, for cases of non-zero values of m, we can form new pairs of real-valued orbitals (What? Are we God? But yes, we are allowed to do this particular limited operation!) with specific directional characteristics by 'linearly combining' the two orbitals with same values of n and l but additive inverse values of m. Thus from the 2p₊₁ and 2p_–1 orbitals, we create the pair of linear combinations (1/2)^1/2.(2p₊₁ + 2p_–1) and (1/2)^1/2.(2p₊₁ – 2p_–1)/i  (here (1/2)^1/2 arises as a normalisation constant). We find that the first one is cylindrically symmetric around the x-axis [it has a functional form N.x. exp{–Zr/(2a_o)}], whereas the second is cylindrically symmetric around the y-axis [having a functional form N.y. exp{–Zr/(2a_o)}]; so we call the first combination as 2p_x and the second one as 2p_y. Similarly from the 3d₊₁ and 3d_–1 orbitals, we would get the combinations (3d₊₁ + 3d_–1)/2^1/2 and (3d₊₁ – 3d_–1)/2^1/2, while from 3d₊₂ and 3d_–2, we get the combinations (1/2)^1/2.(3d₊₂ + 3d_–2) and (1/2)^1/2.(3d₊₂ – 3d_–2)/i. Judging their directional characteristics, we choose to call the four new d-orbitals as 3d_xz, 3d_yz, 3d_x2–y2 & 3d_xy (while the unchanged 3d_o orbital is called also as 3d_z2). Such real-valued and specific-direction orbitals are more convenient for chemical discussions and understanding. Below we tabulate the three real wavefunctions belonging to the 2p subshell.
Note: By calling the 2p_o function as cylindrically symmetric around z-axis, we mean that when one moves away some fixed distance perpendicularly from the axis, from a particular point on the axis, one comes across the same value of the function.

Table 2.4: Some of the Real Wavefunctions Y_nlm(r,q,f)

2.4. Important characteristics of the radial and the angular wavefunctions: 

The radial wavefunction R_nl(r) describes the variation of the complete wavefunction y_nlm with the radial distance r. Now, so, let's see how R_nl varies with r. For all the s-subshells (i.e., for the l = 0 ones), R_nl starts with a high positive value at r = 0 (this is rather shocking to our common sense, indicating that for the s-subshells the probability of the electron to be at or very near the nucleus is high!) and gradually decreases as r increases up to a limit. For the 1s subshell, R_nl (= R₁₀) always remains positive, and continually decreases with increasing r, asymptotically approaching zero as r tends to infinity (Fig. 2.1). However for 2s, R_nl (= R₂₀) becomes negative as r becomes greater than 2a_o/Z (Why? Find out from Table 2.1, also checking that R₂₀ equals 0 at r = 2a_o/Z), attaining a minimum for some value of r, and then increasing in algebraic value (but decreasing in magnitude) as r increasing further, finally asymptotically approaching zero as r tends to infinity (Fig. 2.1). Drawing a not-to-scale, exam-purpose plot of 3s is now easy for you (do try it!) -- R₃₀ would first decrease from a high positive value, reaching zero and decreasing further till it reaches a negative minimum, then increasing, crossing zero on the way again and then reaching a positive maximum, then decreasing continually, finally asymptotically approaching zero (Fig. 2.1). R_nl for the non-s subshells, however starts with a zero value at r = 0 (more pleasing to our common sense), with R_nl increasing as r increases up to a limit, then decreasing from the maximum attained. For 2p and 3d subshells (i.e., for n – l = 1), the said decrease with r is continual throughout, with asymptotic decrease of R_nl towards 0 (Fig. 2.1). However for 3p, 4p, 4d etc. subshells (with n – l > 1), the decrease of R_nl leads to a negative-value minimum followed by increase with further increase of r (Fig. 2.1). [Figures reproduced from Sen*]

 
Fig. 2.1                                                   Fig. 2.2

At the end of this discussion, let us now judge the number of intermediate (i.e., in between r = 0 and r = a) locations having zero value for the radial wavefunction R_nl -- such intermediate locations are called the radial nodes. The number of radial nodes for a radial wavefunction is n – l – 1, isn't it? (Do check it yourself from the above plots). 

As the angular wavefunction Y_lm(q,f) depends on two variables (i.e., q and f), that too angle-variables, similar two-dimensional (i.e., on-paper) plotting of Y_lm versus its variables is unwarranted. Instead we try to visualise their functional variations. Y₀₀ is a constant with no angular variations. Y₁₀ is evidently maximum at q = 0 and minimum at q = p (but magnitude-wise maximum there also), but zero on the plane (i.e., xy plane) with q = p/2 (thus, this plane is a nodal surface for Y₁₀). The effect of the radial nodes and the angular nodal surfaces result in nodal surfaces for the complete wavefunction Y_nlm. Thus for Y₂₀₀ (i.e., 2s orbital), there is one spherical nodal surface with radius 2a_o/Z from the nucleus, whereas for Y₂₀₀ (i.e., 2p_o orbital), there is one planar nodal surface which is the xy (i.e., XOY) plane. It can be shown (cf. Levine) that for each of the real hydrogenlike orbitals, the total number of nodal surfaces is n – 1 (thus, 1s orbital has no nodal surface).

Let us now consider the probability of the electron being observed with the radial distance r lying between some given value r and (r+dr), irrespective of the direction from the nucleus (and so, irrespective of values of the angles q and f). We knew that the probability of the electron lying with r between r & (r+dr), q between q & (q+dq), and f between f & (f+df) is |y(r,q,f)|² r² sinq dq df dr = |R(r)|² |Y(q,f)|² r² sinq dq df dr. So the probability that the electron lies with r between r & (r+dr), irrespective of the values of the angles q and f (or in other words within an annular region between two spheres with radius r and r+dr) is dp_r = ₀∫^2p₀∫^p |R(r)|² |Y(q,f)|² r² sinq dq df dr = |R(r)|² r² dr. ₀∫^2p₀∫^p |Y(q,f)|² sinq dq df. As, we know, ₀∫^2p₀∫^p |Y(q,f)|² sinq dq df always equals 1, so this probability dp_r = |R(r)|² r² dr. However, as the radial wavefunction R(r) is always real (Table 2.1), |R(r)|² further equals R²(r) or R² (also denoted as R_nl²). Thus the probability dp_r = r² R² dr = f(r) dr, where f(r) = r² R² (rhyme-like to remember, isn't it?) is called the radial distribution function (this name implying that the probability distribution along the radial distance r is given by this function). In contrast to the radial wavefunction R_nl(r), the radial distribution function r² R_nl²(r) has zero value at r = 0 (obviously due to the r² factor in it) even for the s-type wavefunctions, have higher values for larger values of r (due to the r² factor), and is everywhere positive-valued (as R_nl(r) function remains squared here) -- check the r² R_nl²(r) vs. r plots in Fig. 2.2. 

 
2.5. Orbital energies in hydrogen atom and in the many-electron atoms: 

In a hydrogen atom or in any other hydrogen-like atomic system with only one electron (e.g., O⁷⁺ ion), the electron's energy (here same as the orbital energy -- of the orbital occupied by the lone electron) is found to be –Z²e²/(8pe_on²a_o) =  –(13.6Z²/n²) eV, also expressed as –(Z²/2n²)E_H -- E_H being the atomic unit of energy also called the Hartree, equalling 27.2 eV. [Putting values of e = 1.602 x 10^–19 C, 1/4pe_o = 8.988 x 10⁹ N m² C^–2 and a_o = 0.529 x 10^–10 m in the first expression, verify that it indeed equals –(2.18Z²/n²) x 10^–18 J = –(13.6 Z²/n²) eV = –(Z²/2n²)E_H. Let you also hear here that a_o, the atomic unit of length, actually equals (4pe_o)h²/(4p²me²), thus approximately equalling (4pe_o)h²/(4p²m_ee²) -- may verify its given value (i.e., 0.529 Å) also.]
Problem: For Li²⁺ ion, what is the wavenumber 1/l of the second line in the Balmer series? Ans: For hydrogenlike systems, the wavenumber expression is, obviously, 1/l =  Z².R_y.(1/n_l² – 1/n_u²). So here, as Z = 3 (Li²⁺ ion), n_l = 2 (Lyman series), n_u = n_l + 2 = 4 (the second line), we get 1/l = 1.851 x 10⁷ m^–1.

However, for a many-electron atom, its orbital energies aren't given by the above expression –(Z²/2n²)E_H. This is because an electron in any given orbital are greatly screened (shielded) from the attraction of the nucleus by the presence of the other electrons, so that the electron considered do not experience the full nuclear charge Z but rather an effective nuclear charge Z_ef, less than Z by an screening constant s (value of s depends on the atom and on the orbital), i.e., Z_ef = Z–s. Thus the orbital energy actually becomes –(Z_ef²/2n²)E_H = –{(Z–s)²/(2n²)}E_H.
Screening or shielding arises because an electron in a many-electron atom is simultaneously acted upon by the nucleus and the other electrons. The nucleus, with its charge +Ze, attracts the electron to itself whereas every other electron, with a charge –e each, repels it. As the electrons in an atom, in an averaged-out consideration, remain spherically symmetrical around the nucleus, the net result of the nuclear attraction and the repulsion by the other electrons is that the considered electron is pulled towards the nucleus with a much weaker force compared to the actual nuclear attraction. This is screening (or the screening effect), as a result of which the electron experiences an effective nuclear charge Z_ef = Z–s instead of the actual nuclear charge Z.
Problem: What are the orbital energies of the 1s, the 2s and the 2p orbitals in an O atom, with the screening constant equalling (as per the well-known Slater's rules) 0.30, 3.45 and 3.45 for the 1s, 2s and 2p orbitals respectively. Ans.: Here Z = 8. For the 1s orbital, n = 1 whereas for the 2s and 2p orbitals n = 2. So, orbital energy for 1s is –(7.7²/2)E_H whereas for 2s and 2p it is –(4.55²/8)E_H (actually the 2s & 2p energies should have been different, but Slater's rules is a rather coarse approximation). Have you noted (verify the values) that the orbital energies are invariably lower for larger-Z atoms e.g., that 2s of O-atom would lie lower than 2s of H-atom? Note also that screening raises the outer-orbital energies more (with larger s values) than the inner-orbital ones.

There's another effect that causes, via the screening effect, unequal variations in orbital energies. The s orbitals penetrate, i.e., have appreciable function values, up to the region near the nucleus, whereas the p and the d orbitals do not similarly penetrate up to the near-nuclear region. Such varying penetration of different orbitals to the inner region of the atom is called the penetration effect. Because of penetration effect, the orbitals with differing values of the azimuthal quantum number l are screened to different extent, thus producing unequal variations in their energies. In particular, the orbital energies of the different subshells of the same atomic shell (e.g., the 3s, 3p and 3d subshell of the 3rd shell -- i.e., the M shell) becomes different due to the combined action of the screening and the penetration effects. As per the hydrogenlike-system orbital-energy expression –(Z²/2n²)E_H, such subshell energies such as the 3s, 3p & 3d energies would have been identical (as Z and n values are same for all of them) -- they are indeed same in a H-atom or in an one-electron atomic ion. However, as per the screening-corrected energy expression –{(Z–s)²/(2n²)}E_H, these subshell energies in a many-electron atom are different, as the screening constant s is different for the different subshells. This difference in screening constant occurs because a lower-l subshell (e.g., 3s) penetrates more into the inner core and thus are screened less (i.e., s is smaller), whereas a higher-l subshell (e.g., 3d) penetrates less into the inner core, with its electrons remaining more in the outer region, and thus are screened more (i.e., s is larger). So for the 3s subshell, the orbital energy (= –{(Z–s)²/18}E_H) is lower* whereas for the 3d subshell the orbital energy (= –{(Z–s)²/18}E_H) is higher (what's this 18?), and for 3p subshell it is in between (thus giving the familiar energy variation 3s < 3p < 3d). For the d-subshells (and obviously also for the f-ones), the subshell energy thus becomes so high that it even becomes more than that of the s-subshell of the next shell -- thus energy of the 3d subshell is greater than that of the 4s, and similar is the case with the 4d and the 5s subshells.
* Note: You may chant, about the 3s subshell, the following rhyme: Smaller s, bigger (Z–s), bigger (Z–s)², lower –{(Z–s)²/18}E_H. For the 3d one, it would be: Bigger s, smaller (Z–s), smaller (Z–s)², higher –{(Z–s)²/18}E_H.
Problem: Invoke the Slater's rules to verify that for Ca atom, the 3d orbital energy is indeed more than that of 4s (so that for Ca atom the ground electronic configuration is [Ne]3s²3p⁶4s² but not [Ne]3s²3p⁶3d²).
 

2.6. The two ways of representing shapes of electronic orbitals: 

Atomic orbitals, and also molecular ones, are made of electronic wavefunctions or of probabilities, and so resemble rather coils of chimney smoke instead of solid dumbbells. Nevertheless, one can discuss their spatial shapes just as one discusses about the shapes of such hazy objects in daily life. There are two popular ways of representing the three-dimensional shapes of atomic orbitals, called the electron-dot representation and the boundary-surface representation.

In the electron-dot representation, the relative probability of observing the electron in a given location is represented by the density of dots in that location. Wherever the electron probability is more, the dots are more dense, and wherever that probability is less, the dots are sparse, as shown in the electron dot representation for the 1s, 2s and 2p_o orbitals (Fig. 2.3).

In the boundary-surface representation of the hydrogenlike orbital wavefunction y(r,q,f), closed surfaces (called boundary surfaces) are drawn that enclose atomic regions with high magnitude of y, and in such a way so that a fixed major part (say, 90%) of the total electron probability gets enclosed within such surfaces. To realistically represent the orbital shapes, they are drawn in a characteristic method, namely by plotting the magnitude |y(r,q,f)| along the radial distance r, assuming a fixed value of the variable r in the function y(r,q,f), versus differing values of the coordinate angles q and f indicating all possible directions. [Note: This is similar to plotting the value of a function f(x,y) along y axis versus differing values of x, assuming a fixed value for the variable y.] This procedure gives the familiar spherical shapes for representations of 1s and 2s orbitals as well as the familiar 'two touching spheres' representations for the 2p_o (or 2p_z), 2p_x & 2p_y orbitals (Fig. 2.4). However, the boundary-surface representation of the orbital probability density |y|² is more popular: in this representation, the square of the magnitude i.e., |y(r,q,f)|² is instead plotted along the radial distance r, versus differing values of q and f. It is this probability-density representation that gives the more familiar 'dumbbell shaped' representations for the three aforesaid real 2p orbitals (Fig. 2.4).

To get a feel of the procedure for obtaining the boundary surface representations, let us attempt the two dimensional, on-paper plots corresponding to the actual (three-dimensional) boundary-surface representations for the 2p_z orbital (as in Table 2.4) of the H-atom (Z = 1). Let the plane of your paper be the YOZ plane with y-axis be the vertical axis therein. So the coordinate q becomes the angle made by the line OP joining any on-paper point P and the origin O, with the rightward horizontal direction (Fig. 2.5). Let us now vary the angles q and f. For the 2p_z orbital, f doesn't have any effect on y values (Table 2.4) and anywhere on the plane of our paper (i.e., the YOZ plane) f must equal ±90° = ±p/2 = ±1.571, so let us vary only q in steps of 10° (i.e., of p/18 radian) from its complete range 0° to 180°, fix the value of the variable r as 4a_o (need to assume a sufficiently high fixed value of variable r, as mentioned in above paragraph, so that a major part of the electron probability gets included within the boundary surface regions obtained thereby), and get the values of |y(r,q,f)| and |y(r,q,f)|² (as in Table 2.5 & Fig. 2.5 in another page) to be plotted along the radial distance. For the |y| plot, here is provided the (z, y) points to plot in a graph or in computer (if you use Microsoft Excel, remember to rearrange the horizontal-axis values in increasing order and to put the negative vertical-axial values in a third column); obtain them yourself for the |y|² plot. Plotting them, you may verify that you indeed get the two 2p_z plots shown in Fig 2.4!