G_tensor questions - units, magnitude, $gauge_origin

rkingsbury · January 7, 2025, 4:24pm

Hello, I am exploring how to do g-tensor calculations in Q-Chem v6.2.2 and have encountered a few questions. I’ll state the questions first and then provide input/output below. Thanks in advance for any help on these!

1. Units of isotropic tensor appear incorrect

When I examine the output g_tensor from the example in the documentation, it appears that the tensor components are given in ppm units but the isotropic and anisotropic values are mis-labeled. They are also stated to be in ppm units but appear to have been divided by 1e6.

1/3 * (85.8 + 13469.8 + 3173.8) = 5576.46, which is exactly 10^6 larger than the value printed. The same error appears to affect the anisotropic tensor output 2 lines below.

------------------------------------------------------------------------------
                                   g-tensor HSO-amplitude response converge

 G-Tensor Spin-Orbit Tensor G_IJ, I,J=X,Y,Z (ppm):
 ------------------------------------------------------------------------------
         85.87798248   -0.00000000    0.00000000
         -0.0000000013469.83548653   -0.00000000
          0.00000000   -0.00000000 3173.77940523
 ------------------------------------------------------------------------------
 Isotropic g-tensor: 1/3 * (g_xx + g_yy + g_zz)
 Isotropic g-tensor in ppm = 0.00557650
 Anisotropic g-tensor: SQRT(0.5*((g_xx-g_yy)^2 + (g_yy-g_zz)^2 + (g_zz-g_xx)^2))
 Anisotropic g-tensor in ppm = 0.01213825
 ------------------------------------------------------------------------------

2. Magnitude of the g-tensor

For a relatively simple spin 1/2 system such as the one in the example, I would expect the electronic g factor to be close to 2 (the value for a single electron), but the value calculated by Q-Chem is ~0.0055. Am misunderstanding what this g-tensor is? If I run the calculation for an even simpler system (a single hydrogen atom), I get a g_tensor of exactly zero, which I also found confusing. I would greatly appreciate any help understanding how to inerpret these magnitudes.

3. Meaning of $gauge_origin$

I could not find the $gauge_origin$ input flag documented anywhere, but it is included in the example. What is it’s purpose?

** Example Input / Output 1: the example from the documentation **

$molecule
1 2
O    0.00000000   0.00000000   0.13475163
H    0.00000000  -1.70748899  -1.06930309
H    0.00000000   1.70748899  -1.06930309
$end

$rem
INPUT_BOHR      = true
METHOD          = ccsd
BASIS           = 3-21g
CC_REF_PROP     = true
G_TENSOR        = true
N_FROZEN_CORE   = 0
NO_REORIENT     = true
SCF_CONVERGENCE = 12
CC_CONVERGENCE  = 12
POINT_GROUP_SYMMETRY = false
$end

$gauge_origin
0.000000 0.000000  0.0172393
$end

------------------------------------------------------------------------------
                                   g-tensor HSO-amplitude response converge

 G-Tensor Spin-Orbit Tensor G_IJ, I,J=X,Y,Z (ppm):
 ------------------------------------------------------------------------------
         85.87798248   -0.00000000    0.00000000
         -0.0000000013469.83548653   -0.00000000
          0.00000000   -0.00000000 3173.77940523
 ------------------------------------------------------------------------------
 Isotropic g-tensor: 1/3 * (g_xx + g_yy + g_zz)
 Isotropic g-tensor in ppm = 0.00557650
 Anisotropic g-tensor: SQRT(0.5*((g_xx-g_yy)^2 + (g_yy-g_zz)^2 + (g_zz-g_xx)^2))
 Anisotropic g-tensor in ppm = 0.01213825
 ------------------------------------------------------------------------------

** Example Input / Output 2: H atom **

$molecule
0 2
H    0.00000000   0.00000000   0.00000000
$end

$rem
METHOD          = ccsd
BASIS           = 3-21g
CC_REF_PROP     = true
G_TENSOR        = true
N_FROZEN_CORE   = 0
NO_REORIENT     = true
SCF_CONVERGENCE = 12
CC_CONVERGENCE  = 12
POINT_GROUP_SYMMETRY = false
$end

$gauge_origin
0.000000 0.000000  0.000000
$end

 ------------------------------------------------------------------------------
                                   g-tensor HSO-amplitude response converge

 G-Tensor Spin-Orbit Tensor G_IJ, I,J=X,Y,Z (ppm):
 ------------------------------------------------------------------------------
          0.00000000    0.00000000    0.00000000
          0.00000000    0.00000000    0.00000000
          0.00000000    0.00000000    0.00000000
 ------------------------------------------------------------------------------
 Isotropic g-tensor: 1/3 * (g_xx + g_yy + g_zz)
 Isotropic g-tensor in ppm = 0.00000000
 Anisotropic g-tensor: SQRT(0.5*((g_xx-g_yy)^2 + (g_yy-g_zz)^2 + (g_zz-g_xx)^2))
 Anisotropic g-tensor in ppm = 0.00000000
 ------------------------------------------------------------------------------

kaushik · January 9, 2025, 7:57am

(1) Yes, it seems the (an)isotropic values are indeed not in ppm unlike the tensor elements. Thanks for informing us. I’ll add a note regarding this feature.

(2) What’s printed are the Delta g values here, i.e., the shifts. Please refer to the related paper: J. Phys. Chem. A 2023, 127, 8459−8472. I’d also recommend that, when in doubt, you run a reported calculation in the reference paper yourself to double check and understand the printed values.

(3) Refer to papers related to CCSD g-tensors for theoretical aspects of this property, e.g., the above paper and papers from Prof. J. Gauss’ group. These properties are gauge dependent, hence this $gauge_origin section for customization. Refer to this section in the Q-Chem User Manual: 7.10.20 Analytic Gradients and Properties for CCSD and EOM-XX-CCSDâ£ 7.10 Coupled-Cluster Excited-State and Open-Shell Methods â£ Chapter 7 Open-Shell and Excited-State Methods â£ Q-Chem 6.2 Userâs Manual.

rkingsbury · January 9, 2025, 1:37pm

Thank you very much @kaushik ; all very helpful!

If I may, I suggest the following enhancements to the Q-Chem manual to clarify these points for others:

Since the equations in Section 10.10.4.3 all define the g tensor as the full tensor (including g_e), adding a statement in the manual (and possibly even the program output) that “The g-tensor is reported as the shift relative to g_e, following the standard convention.” would be helpful
I did not find the paper you referred me to (https://pubs.acs.org/doi/10.1021/acs.jpca.3c04134?articleRef=test) cited anywhere in the associated sections of the Q-Chem manual. Perhaps add a reference in Section 10.10.4.3? You could even use the statement from the paper " Following the common convention, we report shifts in g -tensor values (Δg ) relative to the electron spin g -factor, g e = 2.002319"
For whatever reason, Section 7.10.20 does not appear in results when I search for “gauge_origin” using the search box. (I did so before posting here, hoping to find exactly that description).

jherbert · January 9, 2025, 5:01pm

Try searching the PDF manual, where a search for “gauge_origin” certainly terms up the appropriate section.