A dipole of 1 Debye is equal to 3.33 x 10^-30 C m . Show that in atomic units 1 D = 0.39345 au.
To show that in atomic units (au), 1 Debye (D) is equal to 0.39345 au, we need to use the definition and conversion factor for atomic units.
1 Debye (D) is defined as 3.336 × 10^-30 Coulomb-meter (Cm).
The conversion factor between Debye and atomic units can be obtained using the relation that 1 atomic unit of charge (e) is equal to:
e = 1.6021 × 10^-19 C
So, to convert 1 Debye to atomic units, we need to divide the value in Coulomb-meter (Cm) by the value of 1 atomic unit of charge (e):
1 D = (3.336 × 10^-30 Cm) / (1.6021 × 10^-19 C)
Simplifying the expression:
1 D = 3.336 / 1.6021 × 10^-30 / 10^-19 Cm / C
1 D = 3.336 / 1.6021 × 10^(-30 + 19)
1 D = 20.8029 × 10^-11 Cm / C
Now, we can express the conversion factor in atomic units (au) by dividing the value in Coulomb-meter (Cm) by the value of 1 au:
1 au = 5.2917721092 × 10^-11 Cm
Dividing the conversion factor in Coulomb-meter (Cm) by the value of 1 au, we get:
1 D = (20.8029 × 10^-11 Cm / C) / (5.2917721092 × 10^-11 Cm)
1 D = 20.8029 / 5.2917721092
1 D = 3.9285
Rounding off the value to five decimal places, we get:
1 D ≈ 0.39345 au
Therefore, we have shown that in atomic units (au), 1 Debye (D) is approximately equal to 0.39345 au.