Return to Answer

If you inspect Table II in Ref. 1, which you will find referenced in the link you provide to the WebElements website, you'll see that AU units are used (also in the linked page), not pm, so the radius of the 3d Cu orbital in Ref. 1 is actually

$$ r_{\textrm{Mann}} = 0.613 \times 0.529 Å = 0.324 Å $$

or 32.4 pm.

As for Slaters approach, it is not enough to simply divide $a_0$ by the effective nuclear charge. You can only do this for an unshielded hydrogenic 1s orbital, otherwise it is generally necessary to account for the principal quantum number (in Slater's method, this may be an effective value not equal to the principal quantum number, see Slater's original article$^2$). The location of the maximum follows from the functional form of a Slater wave function:

$$\psi \propto r^{n^*-1}\exp\left({\frac{Z_{\textrm{eff}}}{n^*a_0}r}\right)$$

The effective quantum number is n*=3 for a 3rd shell electron so that

$$r_{\textrm{Slater}} = \frac{n^{*2}}{Z_{\textrm{eff}}}=\frac{9}{13.2}\times0.529 Å = 0.361 Å $$

or 36.1 pm.

The difference in $r_{\textrm{max}}$ obtained by the two approaches is therefore not that dramatically different.

Of course there are two other things that seem to be important here: (1) whether using an accurate value of $r_{\textrm{max}}$ can be expected to provide you with an accurate value of the property you seek and (2) how to obtain an accurate estimate of $r_{\textrm{max}}$for a d-orbital of Cu in the transition metal complex.

References

1. J.B. Mann, Atomic Structure Calculations II. Hartree-Fock wave functions and radial expectation values: hydrogen to lawrencium, LA-3691, Los Alamos Scientific Laboratory, USA, 1968.

2. J.C. Slater. Atomic Shielding Constants. Phys. Rev., 36, 57, 1930.

If you inspect Table II in Ref. 1, which you will find referenced in the link you provide to the WebElements website, you'll see that AU units are used (also in the linked page), not pm, so the radius of the 3d Cu orbital in Ref. 1 is actually

$$ r_{\textrm{Mann}} = 0.613 \times 0.529 Å = 0.324 Å $$

or 32.4 pm.

As for Slaters approach, it is not enough to simply divide $a_0$ by the effective nuclear charge. You can only do this for an unshielded hydrogenic 1s orbital, otherwise it is generally necessary to account for the principal quantum number (in Slater's method, this may be an effective value not equal to the principal quantum number, see Slater's original article$^2$). The location of the maximum follows from the functional form of a Slater wave function:

$$\psi \propto r^{n^*-1}\exp\left({\frac{Z_{\textrm{eff}}}{n^*a_0}r}\right)$$

In general, if you change the nuclear charge of a hydrogenic atom you can perform the transformation $a_0 \rightarrow a_0/Z_{\textrm{eff}}$ to obtain the new orbitals$^3$.

The effective quantum number is n*=3 for a 3rd shell electron so that

$$r_{\textrm{Slater}} = \frac{n^{*2}}{Z_{\textrm{eff}}}=\frac{9}{13.2}\times0.529 Å = 0.361 Å $$

or 36.1 pm.

The difference in $r_{\textrm{max}}$ obtained by the two approaches is therefore not that dramatically different.

Of course there are two other things that seem to be important here: (1) whether using an accurate value of $r_{\textrm{max}}$ can be expected to provide you with an accurate value of the property you seek and (2) how to obtain an accurate estimate of $r_{\textrm{max}}$for a d-orbital of Cu in the transition metal complex.

References

1. J.B. Mann, Atomic Structure Calculations II. Hartree-Fock wave functions and radial expectation values: hydrogen to lawrencium, LA-3691, Los Alamos Scientific Laboratory, USA, 1968.

2. J.C. Slater. Atomic Shielding Constants. Phys. Rev., 36, 57, 1930.

3. W. Greiner. Quantum Mechanics: An Introduction. Springer 1994.

If you inspect Table II in Ref. 1, which you will find referenced in the link you provide to the WebElements website, you'll see that AU units are used (also in the linked page), not pm, so the radius of the 3d Cu orbital in Ref. 1 is actually

$$ r_{\textrm{Mann}} = 0.613 \times 0.529 Å = 0.324 Å $$

or 32.4 pm.

As for Slaters approach, it is not enough to simply divide $a_0$ by the effective nuclear charge. You can only do this for an unshielded hydrogenic 1s orbital, otherwise it is generally necessary to account for the principal quantum number (in Slater's method, this may be an effective value not equal to the principal quantum number, see Slater's original article$^2$). The location of the maximum follows from the functional form of a Slater wave function:

$$\psi \propto r^{n-1}e^{\frac{Z_{\textrm{eff}}}{n}r}$$

The effective quantum number is n*=3 for a 3rd shell electron so that

$$r_{\textrm{Slater}} = \frac{n^{*2}}{Z_{\textrm{eff}}}=\frac{9}{13.2}\times0.529 Å = 0.361 Å $$

or 36.1 pm.

The difference in $r_{\textrm{max}}$ obtained by the two approaches is therefore not that dramatically different.

Of course there are two other things that seem to be important here: (1) whether using an accurate value of $r_{\textrm{max}}$ can be expected to provide you with an accurate value of the property you seek and (2) how to obtain an accurate estimate of $r_{\textrm{max}}$for a d-orbital of Cu in the transition metal complex.

References

1. J.B. Mann, Atomic Structure Calculations II. Hartree-Fock wave functions and radial expectation values: hydrogen to lawrencium, LA-3691, Los Alamos Scientific Laboratory, USA, 1968.

2. J.C. Slater. Atomic Shielding Constants. Phys. Rev., 36, 57, 1930.

If you inspect Table II in Ref. 1, which you will find referenced in the link you provide to the WebElements website, you'll see that AU units are used (also in the linked page), not pm, so the radius of the 3d Cu orbital in Ref. 1 is actually

$$ r_{\textrm{Mann}} = 0.613 \times 0.529 Å = 0.324 Å $$

or 32.4 pm.

As for Slaters approach, it is not enough to simply divide $a_0$ by the effective nuclear charge. You can only do this for an unshielded hydrogenic 1s orbital, otherwise it is generally necessary to account for the principal quantum number (in Slater's method, this may be an effective value not equal to the principal quantum number, see Slater's original article$^2$). The location of the maximum follows from the functional form of a Slater wave function:

$$\psi \propto r^{n^*-1}\exp\left({\frac{Z_{\textrm{eff}}}{n^*a_0}r}\right)$$

The effective quantum number is n*=3 for a 3rd shell electron so that

$$r_{\textrm{Slater}} = \frac{n^{*2}}{Z_{\textrm{eff}}}=\frac{9}{13.2}\times0.529 Å = 0.361 Å $$

or 36.1 pm.

The difference in $r_{\textrm{max}}$ obtained by the two approaches is therefore not that dramatically different.

Of course there are two other things that seem to be important here: (1) whether using an accurate value of $r_{\textrm{max}}$ can be expected to provide you with an accurate value of the property you seek and (2) how to obtain an accurate estimate of $r_{\textrm{max}}$for a d-orbital of Cu in the transition metal complex.

References

1. J.B. Mann, Atomic Structure Calculations II. Hartree-Fock wave functions and radial expectation values: hydrogen to lawrencium, LA-3691, Los Alamos Scientific Laboratory, USA, 1968.

2. J.C. Slater. Atomic Shielding Constants. Phys. Rev., 36, 57, 1930.

If you inspect Table II in Ref. 1, which you will find referenced in the link you provide to the WebElements website, you'll see that AU units are used (also in the linked page), not pm, so the radius of the 3d Cu orbital in Ref. 1 is actually

$$ r_{\textrm{Mann}} = 0.613 \times 0.529 Å = 0.324 Å $$

or 32.4 pm.

As for Slaters approach, it is not enough to simply divide $a_0$ by the effective nuclear charge, as you would for a "naked" hydrogenic system. Instead you are supposed to use an effective principal quantum number, which is n*=3 for a 3rd shell electron (see Slater's original article$^2$), so that

$$r_{\textrm{Slater}} = \frac{n^{*2}}{Z_{\textrm{eff}}}=\frac{9}{13.2}\times0.529 Å = 0.361 Å $$

or 36.1 pm.

The difference in $r_{\textrm{max}}$ obtained by the two approaches is therefore not that dramatically different.

Of course there are two other things that seem to be important here: (1) whether using an accurate value of $r_{\textrm{max}}$ can be expected to provide you with an accurate value of the property you seek and (2) how to obtain an accurate estimate of $r_{\textrm{max}}$for a d-orbital of Cu in the transition metal complex.

References

1. J.B. Mann, Atomic Structure Calculations II. Hartree-Fock wave functions and radial expectation values: hydrogen to lawrencium, LA-3691, Los Alamos Scientific Laboratory, USA, 1968.

2. J.C. Slater. Atomic Shielding Constants. Phys. Rev., 36, 57, 1930.

If you inspect Table II in Ref. 1, which you will find referenced in the link you provide to the WebElements website, you'll see that AU units are used (also in the linked page), not pm, so the radius of the 3d Cu orbital in Ref. 1 is actually

$$ r_{\textrm{Mann}} = 0.613 \times 0.529 Å = 0.324 Å $$

or 32.4 pm.

As for Slaters approach, it is not enough to simply divide $a_0$ by the effective nuclear charge. You can only do this for an unshielded hydrogenic 1s orbital, otherwise it is generally necessary to account for the principal quantum number (in Slater's method, this may be an effective value not equal to the principal quantum number, see Slater's original article$^2$). The location of the maximum follows from the functional form of a Slater wave function:

$$\psi \propto r^{n-1}e^{\frac{Z_{\textrm{eff}}}{n}r}$$

The effective quantum number is n*=3 for a 3rd shell electron so that

$$r_{\textrm{Slater}} = \frac{n^{*2}}{Z_{\textrm{eff}}}=\frac{9}{13.2}\times0.529 Å = 0.361 Å $$

or 36.1 pm.

The difference in $r_{\textrm{max}}$ obtained by the two approaches is therefore not that dramatically different.

Of course there are two other things that seem to be important here: (1) whether using an accurate value of $r_{\textrm{max}}$ can be expected to provide you with an accurate value of the property you seek and (2) how to obtain an accurate estimate of $r_{\textrm{max}}$for a d-orbital of Cu in the transition metal complex.

References

1. J.B. Mann, Atomic Structure Calculations II. Hartree-Fock wave functions and radial expectation values: hydrogen to lawrencium, LA-3691, Los Alamos Scientific Laboratory, USA, 1968.

2. J.C. Slater. Atomic Shielding Constants. Phys. Rev., 36, 57, 1930.