using $exp(\phi(x))$
$g_{ab} = e_{ab} exp(\phi)$
$g^{ab} = (e^{-1})^{ab} exp(-\phi)$
$g_{ab,c} = e_{ab} e^\phi \phi_{,c} = g_{ab} \phi_{,c}$
$\Gamma_{abc} = \frac{1}{2} (g_{ab} \phi_{,c} + g_{ac} \phi_{,b} - g_{bc} \phi_{,a})$
${\Gamma^a}_{bc} = \frac{1}{2} (\delta^a_b \phi_{,c} + \delta^a_c \phi_{,b} - g_{bc} g^{ad} \phi_{,d})$
${\Gamma^a}_{bc,d} = \frac{1}{2} (
\delta^a_b \phi_{,cd}
+ \delta^a_c \phi_{,bd}
- g_{bc} \phi_{,d} g^{ae} \phi_{,e}
- g_{bc} {g^{ae}}_{,d} \phi_{,e}
- g_{bc} g^{ae} \phi_{,ed}
)$
$ = \frac{1}{2} (\delta^a_b \phi_{,cd} + \delta^a_c \phi_{,bd} - g_{bc} g^{ae} \phi_{,ed})$
${\Gamma^a}_{uc} {\Gamma^u}_{bd} = \frac{1}{4}
(\delta^a_u \phi_{,c} + \delta^a_c \phi_{,u} - g_{uc} g^{ae} \phi_{,e})
(\delta^u_b \phi_{,d} + \delta^u_d \phi_{,b} - g_{bd} g^{uf} \phi_{,f})$
$ = \frac{1}{4} (
+ \delta^a_u \phi_{,c} \delta^u_b \phi_{,d}
+ \delta^a_u \phi_{,c} \delta^u_d \phi_{,b}
- \delta^a_u \phi_{,c} g_{bd} g^{uf} \phi_{,f}
+ \delta^a_c \phi_{,u} \delta^u_b \phi_{,d}
+ \delta^a_c \phi_{,u} \delta^u_d \phi_{,b}
- \delta^a_c \phi_{,u} g_{bd} g^{uf} \phi_{,f}
- g_{uc} g^{ae} \phi_{,e} \delta^u_b \phi_{,d}
- g_{uc} g^{ae} \phi_{,e} \delta^u_d \phi_{,b}
+ g_{uc} g^{ae} \phi_{,e} g_{bd} g^{uf} \phi_{,f}
)$
$ = \frac{1}{4} (
+ \delta^a_b \phi_{,c} \phi_{,d}
+ \delta^a_d \phi_{,c} \phi_{,b}
- \phi_{,c} g_{bd} g^{ae} \phi_{,e}
+ \delta^a_c \phi_{,b} \phi_{,d}
+ \delta^a_c \phi_{,d} \phi_{,b}
- \delta^a_c g_{bd} g^{ef} \phi_{,e} \phi_{,f}
- g_{cd} g^{ae} \phi_{,e} \phi_{,b}
+ g_{bd} g^{ae} \phi_{,e} \phi_{,c}
- g_{bc} g^{ae} \phi_{,e} \phi_{,d}
)$
$ = \frac{1}{4} (
\delta^a_b \phi_{,c} \phi_{,d}
+ 2 \delta^a_c \phi_{,b} \phi_{,d}
- \delta^a_c g_{bd} g^{ef} \phi_{,e} \phi_{,f}
+ \delta^a_d \phi_{,c} \phi_{,b}
- g_{bc} g^{ae} \phi_{,e} \phi_{,d}
- g_{cd} g^{ae} \phi_{,e} \phi_{,b}
)$
$R_{ab} = \frac{1}{4} (
+ \delta^c_a \phi_{,c} \phi_{,b}
+ 2 \delta^c_c \phi_{,a} \phi_{,b}
- \delta^c_c g_{ab} g^{ef} \phi_{,e} \phi_{,f}
+ \delta^c_b \phi_{,c} \phi_{,a}
- g_{ac} g^{ce} \phi_{,e} \phi_{,b}
- g_{cb} g^{ce} \phi_{,e} \phi_{,a}
)$
$= 2 \phi_{,a} \phi_{,b} - g_{ab} g^{cd} \phi_{,c} \phi_{,d}$
$R = 2 g^{ab} \phi_{,a} \phi_{,b} - 4 g^{ab} \phi_{,a} \phi_{,b}$
$= -2 g^{ab} \phi_{,a} \phi_{,b}$
$-\frac{R}{2} = g^{ab} \phi_{,a} \phi_{,b}$
$R_{ab} = 2 \phi_{,a} \phi_{,b} + \frac{1}{2} g_{ab} R$
for $R = 0$ this gives $R_{ab} = 2 \phi_{,a} \phi_{,b}$