  Electronic and phononic thermal conductivity are involved in the thermal
conduction for metals and Wiedemann-Franz law is usually employed to predict
them separately. However, Wiedemann-Franz law is shown to be invalid at
intermediate temperatures. Here, to obtain the accurate thermal conductivity
and Lorenz ratio for metals, the momentum relaxation time is used for
electrical conductivity and energy relaxation time for electronic thermal
conductivity. The mode-level first-principles calculation is conducted on two
representative metals copper and aluminum. It is shown that the method can
correctly predict electrical transport coefficients from 6 to 300 K. Also, the
anomalous Lorenz ratio is observed within the present scheme, which has
significant departure from the Sommerfeld value. The calculation scheme can be
expanded to other metallic systems and is valuable in a better understanding of
the electron dynamics and transport properties of metals.
